MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd
MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd
MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd
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<strong>MYSTERIES</strong> <strong>OF</strong> <strong>THE</strong><br />
<strong>EQUILATERAL</strong> <strong>TRIANGLE</strong><br />
Brian J. M c Cartin<br />
Applied Mathematics<br />
Kettering University<br />
<strong>HIKARI</strong> LT D
<strong>HIKARI</strong> LTD<br />
Hikari <strong>Ltd</strong> is a publisher of international scientific journals and books.<br />
www.m-hikari.com<br />
Brian J. McCartin, <strong>MYSTERIES</strong> <strong>OF</strong> <strong>THE</strong> <strong>EQUILATERAL</strong> <strong>TRIANGLE</strong>,<br />
First published 2010.<br />
No part of this publication may be reproduced, stored in a retrieval system,<br />
or transmitted, in any form or by any means, without the prior permission of<br />
the publisher Hikari <strong>Ltd</strong>.<br />
ISBN 978-954-91999-5-6<br />
Copyright c○ 2010 by Brian J. McCartin<br />
Typeset using L ATEX.<br />
Mathematics Subject Classification: 00A08, 00A09, 00A69, 01A05,<br />
01A70, 51M04, 97U40<br />
Keywords: equilateral triangle, history of mathematics, mathematical biography,<br />
recreational mathematics, mathematics competitions, applied mathematics<br />
Published by Hikari <strong>Ltd</strong>
Dedicated to our beloved<br />
Beta Katzenteufel<br />
for completing our equilateral triangle.
Euclid and the Equilateral Triangle<br />
(Elements: Book I, Proposition 1)
Preface v<br />
PREFACE<br />
Welcome to Mysteries of the Equilateral Triangle (MOTET), my collection<br />
of equilateral triangular arcana. While at first sight this might seem an idiosyncratic<br />
choice of subject matter for such a detailed and elaborate study, a<br />
moment’s reflection reveals the worthiness of its selection.<br />
Human beings, “being as they be”, tend to take for granted some of their<br />
greatest discoveries (witness the wheel, fire, language, music,...). In Mathematics,<br />
the once flourishing topic of Triangle Geometry has turned fallow and<br />
fallen out of vogue (although Phil Davis offers us hope that it may be resuscitated<br />
by The Computer [70]). A regrettable casualty of this general decline in<br />
prominence has been the Equilateral Triangle.<br />
Yet, the facts remain that Mathematics resides at the very core of human<br />
civilization, Geometry lies at the structural heart of Mathematics and the<br />
Equilateral Triangle provides one of the marble pillars of Geometry. As such,<br />
it is the express purpose of the present missive, MOTET, to salvage the serious<br />
study of the equilateral triangle from the dustbin of Mathematical History [31].<br />
Like its musical namesake, MOTET is polyphonic by nature and requires<br />
no accompaniment [10]. Instead of being based upon a sacred Latin text,<br />
it rests upon a deep and abiding mathematical tradition of fascination with<br />
the equilateral triangle. The principal component voices are those of mathematical<br />
history, mathematical properties, Applied Mathematics, mathematical<br />
recreations and mathematical competitions, all above a basso ostinato of<br />
mathematical biography.<br />
Chapter 1 surveys the rich history of the equilateral triangle. This will<br />
entail a certain amount of globetrotting as we visit Eastern Europe, Egypt,<br />
Mesopotamia, India, China, Japan, Sub-Saharan Africa, Ancient Greece, Israel,<br />
Western Europe and the United States of America. This sojourn will<br />
bring us into contact with the religious traditions of Hinduism, Buddhism,<br />
Judaism, Christianity and Scientology. We will find the equilateral triangle<br />
present within architecture, sculpture, painting, body armour, basket weaving,<br />
religious icons, alchemy, magic, national flags, games, insects, fruits and vegetables,<br />
music, television programs and, of course, Mathematics itself. N.B.:<br />
Circa 1000 A.D., Gerbert of Aurillac (later Pope Sylvester II) referred to<br />
the equilateral triangle as “mother of all figures” and provided the formula<br />
A ≈ s 2 ·3/7 which estimates its area in terms of the length of its side to within<br />
about 1% ( N. M. Brown, The Abacus and the Cross, Basic, 2010, p. 109).<br />
Chapter 2 explores some of the mathematical properties of the equilateral<br />
triangle. These range from elementary topics such as construction procedures<br />
to quite advanced topics such as packing and covering problems. Old chestnuts<br />
like Morley’s Theorem and Napoleon’s Theorem are to be found here, but so<br />
are more recent rarities such as Blundon’s Inequality and Partridge Tiling.
vi Preface<br />
Many of these plums may be absorbed through light skimming while others<br />
require considerable effort to digest. Caveat emptor: No attempt has been<br />
made either to distinguish between the two types or to segregate them.<br />
In Chapter 3, we take up the place of the equilateral triangle in Applied<br />
Mathematics. Some of the selected applications, such as antenna design and<br />
electrocardiography, are quite conventional while others, such as drilling a<br />
square hole and wrapping chocolates, are decidedly unconventional. I have<br />
based the selection of topics upon my desire to communicate the sheer breadth<br />
of such applications. Thus, the utilization of the equilateral triangle in detecting<br />
gravitational waves, the construction of superconducting gaskets, cartography,<br />
genetics, game theory, voting theory et cetera have all been included.<br />
The subject of Chapter 4 is the role of the equilateral triangle in Recreational<br />
Mathematics. Traditional fare such as dissection puzzles appear on the<br />
menu, but so do more exotic delicacies such as rep-tiles and spidrons. Devotees<br />
of the work of Martin Gardner in this area will instantly recognize my<br />
considerable indebtedness to his writings. Given his extensive contributions to<br />
Recreational Mathematics, this pleasant state of affairs is simply unavoidable.<br />
Chapter 5 contains a collection of olympiad-caliber problems on the equilateral<br />
triangle selected primarily from previous Mathematical Competitions.<br />
No solutions are included but readily available collections containing complete<br />
solutions are cited chapter and verse. Unless otherwise attributed, the source<br />
material for the biographical vignettes of Chapter 6 was drawn from Biographical<br />
Dictionary of Mathematicians [144], MacTutor History of Mathematics<br />
[230] and Wikipedia, The Free Encyclopedia [330]. Finally, we bid adieu to the<br />
equilateral triangle by taking a panoramic view of its many manifestations in<br />
the world about us. Thus, MOTET concludes with a Gallery of Equilateral<br />
Triangles that has been appended and which documents the multifarious and<br />
ubiquitous appearances of the equilateral triangle throughout the natural and<br />
man-made worlds.<br />
I owe a steep debt of gratitude to a succession of highly professional Interlibrary<br />
Loan Coordinators at Kettering University: Joyce Keys, Meg Wickman<br />
and Bruce Deitz. Quite frankly, without their tireless efforts in tracking down<br />
many times sketchy citations, whatever scholarly value may be attached to<br />
the present work would be substantially diminished. Also, I would like to<br />
warmly thank my Teachers: Harlon Phillips, Oved Shisha, Ghasi Verma and<br />
Antony Jameson. Each of them has played a significant role in my mathematical<br />
development and for that I am truly grateful. Once again, my loving wife<br />
Barbara A. (Rowe) McCartin has lent her Mathematical Artistry to the cover<br />
illustration thereby enhancing the appearance of this work.<br />
Brian J. McCartin<br />
Fellow of the Electromagnetics Academy<br />
Editorial Board, Applied Mathematical Sciences
Contents<br />
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v<br />
1 History of the Equilateral Triangle 1<br />
2 Mathematical Properties of the Equilateral Triangle 29<br />
3 Applications of the Equilateral Triangle 78<br />
4 Mathematical Recreations 103<br />
5 Mathematical Competitions 129<br />
6 Biographical Vignettes 148<br />
A Gallery of Equilateral Triangles 188<br />
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199<br />
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221<br />
vii
Chapter 1<br />
History of the Equilateral<br />
Triangle<br />
The most influential Mathematics book ever written is indisputably Euclid’s<br />
The Elements [164]. For two and a half millenia, it has been the Mathematician’s<br />
lodestone of logical precision and geometrical elegance. Its influence<br />
was even felt in the greatest physics book ever written, Newton’s Principia<br />
Mathematica [227]. Even though Newton certainly discovered much of his mechanics<br />
using calculus, he instead presented his results using the geometrical<br />
techniques of Euclid and even the organization of the text was chosen to model<br />
that of The Elements. It should be pointed out, however, that The Elements<br />
has been scathingly criticised by Russell [260] on logical grounds.<br />
As is evident from the Frontispiece, at the very outset of The Elements<br />
(Book I, Proposition I), Euclid considers the construction of an equilateral triangle<br />
upon a given line segment. However, this is far from the first appearance<br />
of the equilateral triangle in human history. Rather, the equilateral triangle<br />
can be found in the very earliest of human settlements.<br />
(a) (b) (c)<br />
Figure 1.1: Mesolithic House at Lepenski Vir (6000 B.C.): (a) Photo of Base<br />
of House. (b) Geometry of Base of House. (c) Sloped Thatched Walls. [289]<br />
1
2 History<br />
Lepenski Vir, located on the banks of the Danube in eastern Serbia, is an<br />
important Mesolithic archeological site [289]. It is believed that the people of<br />
Lepenski Vir represent the descendents of the early European hunter-gatherer<br />
culture from the end of the last Ice Age. Archeological evidence of human<br />
habitation in the surrounding caves dates back to around 20,000 B.C. The<br />
first settlement on the low plateau dates back to 7,000 B.C., a time when the<br />
climate became significantly warmer.<br />
Seven successive settlements have been discovered at Lepenski Vir with<br />
remains of 136 residential and sacral buildings dating from 6,500 B.C. to 5,500<br />
B.C.. As seen in Figure 1.1(a), the base of each of the houses is a circular<br />
sector of exactly 60 ◦ truncated to form a trapezoid. The associated equilateral<br />
triangle is evident in Figure 1.1(b). Figure 1.1(c) presents an artist’s rendering<br />
of how the completed structure likely appeared. The choice of an equilateral<br />
triangular construction principle, as opposed to circular or rectangular, at such<br />
an early stage (“Stone Age”) of human development is really quite remarkable!<br />
Figure 1.2: Snefru’s Bent Pyramid at Dahshur<br />
A fascination with the equilateral triangle may also be traced back to<br />
Pharaonic Egypt. Snefru (2613-2589 B.C.), first pharaoh of the Fourth Dynasty,<br />
built the first non-step pyramid. Known as the Bent Pyramid of Dahshur,<br />
it is shown in Figure 1.2. It is notable in that, although it began with a slant angle<br />
of 60 ◦ (which would have produced an equilateral triangular cross-section),<br />
the base was subsequently enlarged resulting in a slant angle of approximately<br />
54 ◦ (which would have produced equilateral triangular faces) but, due to structural<br />
instability, was altered once again part-way up to a slope of 43 ◦ .<br />
Later pyramids, such as the Great Pyramid of Khufu/Cheops (2551-2528<br />
B.C.) at Giza (see Figure 1.3), were built with a more conservative slope of<br />
approximately 52 ◦ . Although this pyramid is believed to be built based upon<br />
the golden mean [118, pp. 161-163], a construction based upon the equilateral<br />
triangle [115] has been proposed (see Figure 1.4).<br />
Moving to the Fertile Crescent between the Tigres and Euphrates, we en-
History 3<br />
Figure 1.3: Great Pyramid of Khufu<br />
at Giza<br />
Figure 1.5: Old Babylonian Clay<br />
Tablet BM 15285 [259]<br />
Figure 1.4: Great Pyramid Cross-<br />
Section [115]<br />
Figure 1.6: Iron Haematite Babylonian<br />
Cylinder [206]
4 History<br />
counter the sister-states of Babylonia and Assyria who competed for dominance<br />
over what is now Iraq. Babylonia was a land of merchants and agriculturists<br />
presided over by a priesthood. Assyria was an organized military power ruled<br />
by an autocratic king.<br />
Old Babylonian clay tablet BM 15285 is a mathematical “textbook” from<br />
southern Mesopotamia dating back to the early second-millenium B.C. The<br />
portion shown in Figure 1.5 contains a problem which is believed to have<br />
involved an approximate construction of the equilateral triangle [259, pp. 198-<br />
199]. The iron haematite Babylonian cylinder of Figure 1.6 prominently displays<br />
an equilateral triangle as what is believed to be a symbol of a sacred<br />
Trinity [206, p. 605]. Such cylinders were usually engraved with sacred figures,<br />
accompanied by a short inscription in Babylonian cuneiform characters,<br />
containing the names of the owner of the seal and of the divinity under whose<br />
protection he had placed himself.<br />
Figure 1.7: Asshur-izir-pal [250] Figure 1.8: Triangular Altar [250]<br />
Figure 1.7 shows a stele of the Assyrian King Asshur-izir-pal excavated at<br />
Nimrud [250, p. 97]. In front of this figure, marking the object of its erection, is<br />
an equilateral triangular altar with a circular top. Here were laid the offerings<br />
to the divine monarch by his subjects upon visiting his temple. Figure 1.8<br />
shows another triangular Assyrian altar excavated at Khorsabad [250, p. 273].<br />
Moving further East, we encounter Hinduism which is the largest and indigenous<br />
religious tradition of India [157]. The name India itself is derived<br />
from the Greek Indus which is further derived from the Old Persian Hindu.
History 5<br />
Figure 1.9: Kali Yantra Figure 1.10: Chennai Temple [201]<br />
A yantra is a Hindu mystical diagram composed of interlocking geometrical<br />
shapes, typically triangles, used in meditation when divine energy is invoked<br />
into the yantra by special prayers. Figure 1.9 shows the yantra of Kali (Kalika),<br />
Hindu goddess associated with eternal energy. It is composed of five<br />
concentric downward-pointing equilateral triangles surrounded by a circular<br />
arrangement of eight lotus flowers. The five equilateral triangles symbolize<br />
both the five senses and the five tattvas (air, earth, fire, water and spirit).<br />
The ancient Marundheeswarar Temple of Lord Shiva in Thiruvanmayur,<br />
South Chennai has a series of pillars with beautiful geometric designs and<br />
mathematical motifs [201]. Figure 1.10 displays a pattern of three overlapping<br />
equilateral triangles surrounding a central four-petalled flower. These three<br />
equilateral triangles are connected by a Brunnian link in that no two of them<br />
are linked together but the three are collectively linked; if one of the triangles<br />
is removed then the other two fall apart. These pillars are located near the<br />
sanctum of the goddess Tripurasundari, “belle of the three cities”. The implied<br />
triad of this motif is pregnant with symbolism. The archetypal mantra Aum<br />
has three parts; the yogi’s three principal nadis (bodily energy channels) of<br />
the ida, the pengala and the sushumna form a core tantric triad; the three<br />
sakthis (powers) derived from the goddess are the iccha (desire), the gnana<br />
(knowledge) and the kriya (action); and, of course, there is the triad of Brahma<br />
(The Creator), Vishnu (The Preserver) and Shiva (The Destroyer).<br />
Reaching the Orient, in traditional Chinese architecture, windows were
6 History<br />
Figure 1.11: Chinese Window Lattices [91]<br />
made of a decorative wooden lattice with a sheet of rice paper glued to the inside<br />
in order to block the draft while letting in the light. Iconographic evidence<br />
traces such lattices all the way back to 1000 B.C. during the Chou Dynasty.<br />
They reached their full development during the Qing Dynasty beginning in<br />
1644 A.D. when they began to display a variety of geometrical patterns [217].<br />
Figure 1.11 displays two such examples with an equilateral triangular motif<br />
[91].<br />
The Three Pagodas of Chongsheng Monastery are an ensemble of three<br />
independent pagodas arranged at the vertices of an equilateral triangle located<br />
near the town of Dali in the Yunnan Province of China (Figure 1.12). Unique<br />
in China, legend has it that the Three Pagodas were built to deter natural<br />
disasters created by dragons [333]. The main pagoda, known as Qianxun<br />
Pagoda, was built during 824-840 A.D. by King Quan Fengyou of the Nanzhao<br />
state. Standing at 227 feet, it is one of the highest pagodas of the Tang Dynasty<br />
(618-907 A.D.). This central pagoda is square-shaped and is composed of<br />
sixteen stories, each story with multiple tiers of upturned eaves. There is a<br />
carved shrine containing a white marble sitting Buddha statue at the center<br />
of each facade of every story. The body of the pagoda is hollow from the<br />
first to the eighth story and was used to store sculptures and documents. The<br />
other two sibling pagodas, built around one hundred years later, stand to<br />
the northwest and southwest of Qianxun Pagoda. They are both solid and<br />
octagonal with ten stories and stand to a height of 140 feet. The center of<br />
each side of every story is decorated with a shrine containing a Buddha statue.<br />
During the Edo period (1603-1867), when Japan isolated itself from the<br />
western world, the country developed a traditional Mathematics known as
History 7<br />
Figure 1.12: The Three Pagodas (900 A.D.)<br />
Figure 1.13: Japanese Mathematics:<br />
Wasan [299]<br />
Figure 1.14: Japanese Chainmail:<br />
Hana-Gusari [329]
8 History<br />
Wasan which was usually stated in the form of problems, eventually appearing<br />
in books which were either handwritten with a brush or printed from wood<br />
blocks. The problems were written in a language called Kabun, based on<br />
Chinese, which cannot be readily understood by modern Japanese readers<br />
even though written Japanese makes extensive use of Chinese characters [111].<br />
One such problem due to Tumugu Sakuma (1819-1896) is illustrated in Figure<br />
1.13. The sides AB, BC and CA of an equilateral triangle ABC pass through<br />
three vertices O, L and M of a square. Denoting the lengths of OA, LB and<br />
MC by a, b and c, respectively, show that a = ( √ 3 − 1)(b + c) [299]. F.<br />
Suzuki has subsequently generalized this problem by replacing the square by<br />
an isosceles triangle [300]. (See Recreation 27: Sangaku Geometry.)<br />
Chainmail is a type of armour consisting of small metal rings linked together<br />
in a pattern to form a mesh. The Japanese used mail (kusari) beginning in<br />
the Nambokucho period (1336-1392). Two primary weave patterns were used:<br />
a square pattern (so gusari) and a hexagonal pattern (hana gusari). The base<br />
pattern of hana gusari is a six-link equilateral triangle (Figure 1.14) thereby<br />
permitting polygonal patterns such as triangles, diamonds and hexagons [329].<br />
These were never used for strictly mail shirts, but were instead used over<br />
padded steel plates or to connect steel plates. The resulting mail armour<br />
provided an effective defense against slashing blows by an edged weapon and<br />
penetration by thrusting and piercing weapons.<br />
Figure 1.15: African Eheleo Funnel [140]<br />
The equilateral triangle appears in many other human cultures. In order<br />
to illustrate this, let us return for a moment to the African continent. A<br />
pyramidal basket, known as eheleo in the Makhuwa language, is woven in
History 9<br />
regions of Sub-Saharan Africa such as the North of Mozambique, the South<br />
of Tanzania, the Congo/Zaire region and Senegal [140]. In Mozambique and<br />
Tanzania, it is used as a funnel in the production of salt. As seen in Figure<br />
1.15, the funnel is hung on a wooden skeleton and earth containing salt is<br />
inserted. A bowl is placed beneath the funnel, hot water is poured over the<br />
earth in the funnel, saltwater is caught in the bowl and, after evaporation, salt<br />
remains in the bowl. The eheleo basket has the form of an inverted triangular<br />
pyramid with an equilateral triangular base and isosceles right triangles as its<br />
three remaining faces.<br />
Figure 1.16: Parthenon and Equilateral Triangles [54]<br />
We turn next to the cradle of Western Civilization: Ancient Greece. Figure<br />
1.16 shows the portico of the Parthenon together with superimposed concentric<br />
equilateral triangles, each successive triangle diminished in size by one-half<br />
[54]. This diagram “explains” the incscribing rectangle, the position of the<br />
main cornice, the underside of the architrave, and the distance between the<br />
central columns.<br />
The Pythagorean Tetraktys is shown in Figure 1.17, which is from Robert<br />
Fludd’s Philosophia Sacra (1626), where the image shows how the original<br />
absolute darkness preceded the Monad (1), the first created light; the Dyad (2)<br />
is the polarity of light (Lux) and darkness (Tenebrae), with which the Humid<br />
Spirit (Aqua) makes a third; the combination of the four elements (Ignis, Aer,<br />
Aqua, Terra) provides the foundation of the world. To the Pythagoreans, the<br />
first row represented zero-dimensions (a point), the second row one-dimension<br />
(a line defined by two points), the third row two-dimensions (a plane defined by<br />
a triangle of three points) and the fourth row three-dimensions (a tetrahedron
10 History<br />
Figure 1.17:<br />
Pythagorean Tetraktys<br />
Figure 1.18: Neopythagorean<br />
Tetraktys<br />
Figure 1.19: Triangular<br />
Numbers<br />
defined by four points). Together, they symbolized the four elements: earth,<br />
air, fire and water. The tetraktys (four) was seen to be the sacred decad (ten)<br />
in disguise (1+2+3+4=10). It also embodies the four main Greek musical<br />
harmonies: the fourth (4:3), the fifth (3:2), the octave (2:1) and the double<br />
octave (4:1) [148].<br />
For neophythagoreans [276], the tetraktys’ three corner dots guard a hexagon<br />
(6, symbolizing life) and the hexagon circumscribes a mystic hexagram (two<br />
overlapping equilateral triangles, upward-pointing for male and downwardpointing<br />
for female, denoting divine balance) enclosing a lone dot (Figure 1.18).<br />
This dot represents Athene, goddess of wisdom, and symbolizes health, light<br />
and intelligence. The tetraktys is also the geometric representation of the<br />
fourth of the triangular numbers ∆n = n(n+1)<br />
2 (Figure 1.19).<br />
Figure 1.20: Hebrew<br />
Tetragrammaton<br />
Figure 1.21: Archbishop’s<br />
Coat of Arms<br />
Figure 1.22: Tarot<br />
Card Spread<br />
The tetraktys has also been passed down to us in the Hebrew Tetragrammaton<br />
of the Kabbalah (Figure 1.20) and the Roman Catholic archbishop’s coat<br />
of arms (Figure 1.21). The tetraktys is also used in one Tarot card reading arrangement<br />
(Figure 1.22). The various positions provide a basis for forecasting<br />
future events.
History 11<br />
Figure 1.23: Platonic Triangles: (a) Earth. (b) Water. (c) Air. (d) Fire. [176]<br />
In his Timaeus, Plato symbolizes ideal or real Earth as an Equilateral Triangle<br />
[176, pp. 16-17] (Figure 1.23). A Platonic (regular) solid is a convex<br />
polyhedron whose faces are congruent regular polygons with the same number<br />
of faces meeting at each vertex. Thus, all edges, vertices and angles are congruent.<br />
Three of the five Platonic solids have all equilateral triangular faces<br />
(Figure 1.24): the tetrahedron (1), octahedron (3) and icosahedron (4) [66].<br />
Of the five, only the cube (2) can fill space. The isosahedron, with its twenty<br />
equilateral triangular faces meeting in fives at its twelve vertices, forms the<br />
logo of the Mathematical Association of America. By studying the Platonic<br />
solids, Descartes discovered the polyhedral formula P = 2F + 2V − 4 where P<br />
is the number of plane angles, F is the number of faces, and V is the number<br />
of vertices [3]. Euler independently introduced the number of edges E and<br />
wrote his formula as E = F + V − 2 [252].<br />
Figure 1.24, which was derived from drawings by Leonardo da Vinci, has<br />
an interesting history [44]. Because Leonardo was born on the wrong side<br />
of the blanket, so to speak, he was denied a university education and was<br />
thus “unlettered”. This meant that he did not read Latin so that most formal<br />
learning was inaccessible to him. As a result, when he was middle-aged,<br />
Leonardo embarked on an ambitious program of self-education that included<br />
teaching himself Latin. While in the employ of Ludovico Sforza, Duke of Milan,<br />
he made the fortuitous acquaintance of the famous mathematician Fra<br />
Luca Pacioli. Pacioli guided his thorough study of the Latin edition of Euclid’s<br />
Elements thereby opening a new world of exploration for Leonardo. Soon<br />
after, Leonardo and Fra Luca decided to collaborate on a book. Written by<br />
Pacioli and illustrated by Leonardo, De Divina Proportione (1509) contains an<br />
extensive review of proportion in architecture and anatomy, in particular the
12 History<br />
Figure 1.24: Five Platonic Solids
History 13<br />
golden section, as well as detailed discussions of the Platonic solids. It contains<br />
more than sixty illustrations by Leonardo including the skeletal forms of the<br />
Platonic solids of Figure 1.24. It is notable that this is the only collection of<br />
Leonardo’s drawings that was published during his lifetime. Remarkably, this<br />
work was completed contemporaneously with his magnificent The Last Supper.<br />
Figure 1.25: Thirteen Archimedean Solids<br />
An Archimedean (semiregular) solid is a convex polyhedron composed of<br />
two or more regular polygons meeting in identical vertices. Nine of the thirteen<br />
Archimedean solids have some equilateral triangular faces (Figure 1.25):<br />
truncated cube (1), truncated tetrahedron (2), truncated dodecahedron (3),<br />
cuboctahedron (8), icosidodecahedron (9), (small) rhombicuboctahedron (10),<br />
(small) rhombicosidodecahedron (11), snub cube (12) and snub dodecahedron<br />
(13) [66]. Of the thirteen, only the truncated octahedron (5) can fill space.<br />
Archimedes’ work on the semiregular solids has been lost to us and we only<br />
know of it through the later writings of Pappus.
14 History<br />
Figure 1.26: Eight Convex Deltahedra<br />
Figure 1.27: Two Views of Almost-Convex 18-Sided “Deceptahedron” [134]<br />
Figure 1.28: Non-Convex 8-Sided Deltahedron [134]
History 15<br />
A deltahedron is a polyhedron whose faces are congruent non-coplanar<br />
equilateral triangles [322]. Although there are infinitely many deltahedra,<br />
Freudenthal and van der Waerden showed in 1947 that only eight of them<br />
are convex, having 4 (tetrahedron), 6 (triangular dipyramid), 8 (octahedron),<br />
10 (pentagonal dipyramid), 12 (dodecadeltahedron), 14 (tetracaidecadeltahedron),<br />
16 (heccaidecadeltahedron) and 20 (icosahedron) faces (Figure 1.26)<br />
[66]. These, together with the cube and dodecahedron, bring the number of<br />
convex polyhedra with congruent regular faces up to ten. The absence of an<br />
18-sided convex deltahedron is most peculiar. Figure 1.27 displays two views<br />
of an 18-sided deltahedron due to E. Frost that is so close to being convex<br />
that W. McGovern has named it a “deceptahedron” [134]. In addition to the<br />
octahedron, there is the non-convex 8-sided deltahedron shown in Figure 1.28<br />
[134]. Whereas the regular octahedron has four edges meeting at each of its six<br />
vertices, this solid possesses two vertices where three edges meet, two vertices<br />
where four edges meet and two vertices where five edges meet.<br />
Figure 1.29: The Three Regular Tilings of the Plane<br />
One of the three regular tilings of the plane is comprised of equilateral<br />
triangles (Figure 1.29). Six of the eight semiregular tilings of the plane have<br />
equilateral triangular components (Figure 1.30). Study of such tilings was<br />
inaugurated in the Harmonice Mundi of Johannes Kepler [156].<br />
Figure 1.30: The Eight Semiregular Tilings of the Plane
16 History<br />
(a) (b) (c)<br />
Figure 1.31: Hexagram: (a) Magic Hexagram. (b) Star of David. (c) Chinese<br />
Checkers<br />
A hexagram is a six-pointed star, with a regular hexagon at its center,<br />
formed by combining two equilateral triangles (Figure 1.31). Throughout the<br />
ages and across cultures, it has been one of the most potent symbols used in<br />
magic [49]. The yantra of Vishnu, the Supreme God of the Vishnavite tradition<br />
of Hinduism, contains such a hexagram [193]. Mathematically, a normal magic<br />
hexgram (Figure 1.31(a)) arranges the first 12 positive integers at the vertices<br />
and intersections in such a way that the four numbers on each line sum to the<br />
magic constant M = 26 [87, p. 145]. This can be generalized to a normal<br />
magic star which is an n-pointed star with an arrangement of the consecutive<br />
integers 1 thru 2n summing to a magic number of M = 4n + 2 [308].<br />
The Star of David (Figure 1.31(b)) is today generally recognized as a symbol<br />
of Jewish identity. It is identified with the Shield of David in Kabbalah,<br />
the school of thought associated with the mystical aspect of Rabbinic Judaism.<br />
Named after King David of ancient Israel, it first became associated with the<br />
Jews in the 17th Century when the Jewish quarter of Vienna was formally<br />
distinguished from the rest of the city by a boundary stone having a hexagram<br />
on one side and a cross on the other. After the Dreyfus affair in 19th Century<br />
France, it became internationally associated with the Zionist movement. With<br />
the establishment of the State of Israel in 1948, the Star of David became<br />
emblazoned on the Flag of Israel [278].<br />
Figure 1.31(c) contains the playing board for the inappropriately named<br />
Chinese Checkers. The game was invented not in ancient China but in Germany<br />
by Ravensburger in 1893 under the name “Stern-Halma” as a variation<br />
of the older American game of Halma. “Stern” is German for star and refers<br />
to the star-shaped board in contrast to the square board of Halma. In the<br />
United States, J. Pressman & Co. marketed the game as “Hop Ching Checkers”<br />
in 1928 but quickly changed the name to “Chinese Checkers” as it gained<br />
popularity. This was subsequently introduced to China by the Japanese [236].
History 17<br />
Figure 1.32: The Last Supper (Da Vinci: 1498)<br />
Figure 1.33: Supper at Emmaus<br />
(Pontormo: 1525)<br />
Figure 1.34: Christ of Saint John of<br />
the Cross (Dali: 1951)
18 History<br />
The equilateral triangle is a recurring motif in Christian Art [222]. Front<br />
and center in this genre is occupied by Leonardo da Vinci’s The Last Supper<br />
(see Figure 1.32). Begun in 1495 and finished in 1498, it was painted on<br />
the rear wall of the Refectory at the Convent of Santa Maria delle Grazie.<br />
This mural began to deteriorate in Leonardo’s own lifetime. Its most recent<br />
restoration took twenty years and was only completed in 1999. In this great<br />
masterpiece, the body of Jesus is a nearly perfect equilateral triangle symbolizing<br />
the Trinity. The serene calm of this sacred figure anchors the utter chaos<br />
which has been unleashed by His announcement of the upcoming betrayal by<br />
one of the attending apostles. The theme of Trinity is further underscored by<br />
Leonardo’s partitioning of the apostles into four groups of threes.<br />
The role of the equilateral triangle is even more explicit in Jacopo Pontormo’s<br />
1525 painting Supper at Emmaus (see Figure 1.33). Not only is the<br />
figure of Jesus an equilateral triangle but a radiant triangle with a single eye<br />
hovers above Christ’s head. This symbolizes the all-seeing Eye of God with<br />
the triangle itself representing the Holy Trinity of God the Father, God the<br />
Son and God the Holy Spirit. This painting portrays the occasion of the first<br />
appearance of Christ to two disciples after His Resurrection.<br />
In Salvador Dali’s 1951 Christ of Saint John of the Cross (see Figure 1.34),<br />
the hands and feet of Our Lord form an equilateral triangle symbolizing Father,<br />
Son and Holy Spirit. It depicts Jesus Christ on the cross in a darkened sky<br />
floating over a body of water complete with a boat and fishermen. It is devoid<br />
of nails, blood and crown of thorns because Dali was convinced by a dream<br />
that these features would mar his depiction of the Saviour. This same dream<br />
suggested the extreme angle of view as that of the Father. The name of the<br />
painting derives from its basis in a drawing by the 16th Century Spanish friar<br />
Saint John of the Cross.<br />
The equilateral triangle was frequently used in Gothic architectural design<br />
[30]. Figure 1.35 presents a transveral section of the elevation of the Cathedral<br />
(Duomo) of Milan drawn by Caesare Caesariano and published in his 1521<br />
Italian translation of Vitruvius’ De Architectura. Caesariano was a student of<br />
da Vinci and one of the many architects who produced designs for the Milan<br />
Cathedral over the nearly six centuries of its construction from 1386 to 1965.<br />
Even though this design was ultimately abandoned, it is significant in that<br />
it is one of the rare extant plans for a Gothic cathedral. It clearly shows<br />
the application of ad triangulum design which employs a lattice of equilateral<br />
triangles to control placement of key features and proportions of components.<br />
This technique is combined with one utilizing a system of concentric circles<br />
[285]. It is clear that the equilateral triangle was an important, although by no<br />
means the only, geometric design element employed in the the construction of<br />
the great Gothic cathedrals. This ad triangulum design principle was adopted<br />
by Renaissance artists, particularly in their sacred paintings.
History 19<br />
Figure 1.35: Milan Cathedral (Caesariano: 1521) [142]
20 History<br />
Figure 1.36: Gothic Mason’s Marks [142]<br />
In the Gothic Masons Guilds [142], the Companion (second degree of initiation)<br />
received a personal mason’s mark at the end of his probationary period.<br />
This seal, which would be his sign or password for the remainder of his life,<br />
was not secret and would be used to identify his work and to gain admission<br />
when visiting other lodges. Many of the mason’s marks used by these master<br />
builders to identify themselves and their work were based upon the equilateral<br />
triangle [142, pp. 119-123] (Figure 1.36).<br />
Figure 1.37: Heraldic Cross of the Knights Templar<br />
The Order of the Temple (The Knights Templar) was organized in France<br />
at the commencement of the First Crusade in 1096 A.D. [15]. They trained<br />
like modern day commandos and battled to the death. The Knights Templar<br />
battled like demons for hundreds of years throughout the various Crusades,<br />
meeting their end at Acre, their last stronghold in the Holy Lands, in 1291.<br />
Most of them were butchered by the Moslems and the survivors made their<br />
way to France where their Order was eventually suppressed by the Catholic<br />
Church. The remaining Knights Templar became affiliated with Freemasonry.<br />
The Heraldic Cross of the Knights Templar (a.k.a. cross formée, Tatzenkreuz,<br />
Iron cross, Maltese cross, Victoria cross) is comprised of an arrangement of<br />
four equilateral triangles joined at a common vertex (Figure 1.37).
History 21<br />
Figure 1.38: Masonic Royal Arch Jewel<br />
Freemasonry is a fraternal organization that arose in the late 16th to early<br />
17th Centuries [174]. Freemasonry uses metaphors of stonemasons’ tools and<br />
implements to convey a system of beliefs based upon charitable work, moral<br />
uprightness and fraternal friendship. It has been described as a “society with<br />
secrets” rather than a secret society. The private aspects of Freemasonry concern<br />
the modes of recognition amongst members and particular elements within<br />
their rituals. Both Wolfgang Amadeus Mozart and George Washington were<br />
prominent Freemasons and Masonic Lodges still exist today. The equilateral<br />
triangle is used in Freemasonry as the symbol of the Grand Architect of the<br />
Universe [29] (Figure 1.38).<br />
Figure 1.39: Rosicrucian Cross<br />
Rosicrucianism (The Brotherhood of the Rosy Cross), on the other hand,<br />
was a truly secret society which arose roughly contemporaneously in Germany<br />
in the early 17th Century [50]. At times referred to as the College of Invisibles,<br />
this secret brotherhood of alchemists and sages sought to transform the arts,<br />
sciences, religion, and political and intellectual landscape of Europe. It is<br />
believed likely that both Kepler and Descartes were Rosicrucians [3]. There are<br />
several modern day groups that have styled themselves after the Rosicrucians.<br />
The equilateral triangle is also used in the symbolism of the Rosicrucians<br />
(Figure 1.39).
22 History<br />
Figure 1.40: Vesica Piscis<br />
The vesica piscis (“fish’s bladder”) [204] (Figure 1.40) was the central diagram<br />
of Sacred Geometry for the Christian mysticism of the Middle Ages.<br />
It was the major thematic source for the Gothic cathedrals such as that<br />
at Chartres. Renaissance artists frequently surrounded images of Jesus and<br />
framed depictions of the Virgin Mary with it [285]. Its intimate connection<br />
with √ 3 is revealed by the presence of the equilateral triangles in Figure 1.40.<br />
It in fact predates Christianity [170] and is known in India as mandorla (“almond”).<br />
It was used in early Mesopotamia, Africa and Asia as a symbol of<br />
fertility. To the Pythagoreans, it symbolized the passage of birth. Figure 1.41<br />
shows the image of Jesus Christ enclosed in a vesica piscis from a medieval<br />
illuminated manuscript. Such images allude to his life as a “fisher of men”.<br />
Figure 1.41: Jesus Christ<br />
Utilizing a vesica piscis and a large triangle within it that is further subdivided,<br />
Figure 1.42 reveals the tripartite structure of many naturally occurring<br />
objects [271].
History 23<br />
Figure 1.42: Three-Part Harmony [271]
24 History<br />
Figure 1.43: Alchemical Symbols<br />
Alchemy is an ancient practice concerned with changing base metals into<br />
gold, prolonging life and achieving ultimate wisdom. In particular, alchemy is<br />
the predecessor of modern inorganic chemistry [154]. Alchemy has been widely<br />
practiced for at least 5,000 years in ancient Egypt, Mesopotamia, India, Persia,<br />
China, Japan, Korea, the classical Greco-Roman world, the medieval Islamic<br />
world, and medieval Europe up to the 17th Century. (Isaac Newton devoted<br />
considerably more of his writing to alchemy than to physics and Mathematics<br />
combined! [328]) The alchemical symbols for the four elements (Earth, Wind,<br />
Air and Fire) are all composed of equilateral triangles (Figure 1.43).<br />
Figure 1.44: Scientology Symbol<br />
Fascination with the equilateral triangle persists to this day. The Church<br />
of Scientology [207] was founded by science fiction author L. Ron Hubbard in<br />
1953 based upon his principles of Dianetics [187]. It counts among its adherents<br />
the movie stars John Travolta and Tom Cruise as well as the musicians Chick<br />
Corea and Edgar Winter. The Scientology Symbol is the letter S together<br />
with two equilateral triangles (Figure 1.44). The upper (KRC) triangle symbolizes<br />
knowledge, responsibility and control while the lower (ARC) triangle<br />
symbolizes affinity, reality and communication.
History 25<br />
Figure 1.45: Augmented Triad [218]<br />
An equilateral triangle on the circle-of-fifths [218] produces a musical chord<br />
known as an augmented triad (Figure 1.45). Unlike other kinds of triads, such<br />
as major, minor and diminished, it does not naturally arise in a diatonic scale.<br />
However, the augmented triad does occur in the tonal music of Classical composers<br />
such as Haydn, Beethoven and Brahms. The augmented triad has also<br />
been used by Romantic composers such as Liszt and Wagner to suspend tonality.<br />
Schubert lead the way in organizing many pieces, such as his Wanderer<br />
Fantasy, by descending major thirds which is the component interval of the<br />
augmented triad. Jazz musicians, such as Miles Davis and John Coltrane,<br />
have freely used chord progressions utilizing downward root movement by major<br />
thirds as a substitute for the traditional ii − V − I progression.<br />
(a)<br />
Figure 1.46: Equilateral Triangular Sculpture: (a) Intuition by J. Robinson.<br />
(b) H. S. M. Coxeter holding a model of G. Odom’s 4-Triangle Sculpture. [61].<br />
(b)
26 History<br />
Modern sculpture has not been immune to the influence of the equilateral<br />
triangle. A hollow triangle is defined as the planar region between two<br />
homothetic and concentric equilateral triangles, i.e. a flat triangular ring [61].<br />
Intuition by Australian sculptor John Robinson (Figure 1.46(a)) is composed<br />
of three such triangular rings forming a structure in which certain points<br />
on two outer edges of each ring fit into two inner corners of the next, in cyclic<br />
order. The topology of the assembly is that of the “Borromean rings” (all<br />
three triangles are linked, but no pair is linked), and its symmetry group is<br />
C3, cyclically permuting the three hollow triangles. H. Burgiel et al. [37] have<br />
shown that, for the structure to be realizable in three dimensions, the ratio<br />
of the edge length of the outer triangle to the inner triangle must lie strictly<br />
between one and two. In fact, Robinson utilized a ratio of (2 √ 6+1)/3 ≈ 1.966.<br />
Independently, American artist George Odom assembled four such triangular<br />
rings (with edge ratio 2:1) to form a rigid structure in which the midpoints<br />
of the three outer edges of each ring fit into inner corners of the three remaining<br />
rings (Figure 1.46(b)). The four rings are mutually interlocked and the<br />
symmetry group is the octahedral group O or S4: 24 rotations permuting the<br />
4 hollow triangles in all of the 4! possible ways.<br />
Figure 1.47: Gateway Arch<br />
As an example of the equilateral triangle in contemporary architecture,<br />
consider the Gateway Arch shown in Figure 1.47. (For other examples see<br />
the Gallery of Equilateral Triangles in Appendix A.) It is part of the Jefferson<br />
National Expansion Memorial on the Saint Louis riverfront. Each leg is<br />
an equilateral triangle with sides 54 feet long at ground level and tapering<br />
to 17 feet at the top. The stainless-steel-faced Arch spans 630 feet between<br />
the outer faces of its equilateral triangular legs at ground level and its top
History 27<br />
soars 630 feet into the sky. The interior of the Arch is hollow and contains<br />
a tram transport system leading to an observation deck at the top. This is<br />
complemented by two emergency stairwells of 1076 steps each. The Arch has<br />
no real structural skeleton. Its inner and outer steel skins are joined to form a<br />
composite structure which provides its strength and permanence.<br />
(a) (b) (c)<br />
Figure 1.48: Star Trek: (a) Planet Triskelion in Trinary Star System M24α.<br />
(b) Triskelion Battlefield. (c) Kirk confronts the Providers.<br />
Pop culture has also felt the sway of the equilateral triangle. In the 1960’s<br />
cult classic Star Trek, the 1968 episode The Gamesters of Triskelion (Figure<br />
1.48) sees Captain Kirk, Lieutenant Uhura and Ensign Chekov beamed to the<br />
planet Triskelion which orbits one of a trinary star system (a). There they<br />
must do battle with combatants kidnapped from other worlds on a hexagonal<br />
battlefield with three spiral arms surrounding a central equilateral triangle (b);<br />
all this to satisfy the yearn for wagering of the three disembodied Providers<br />
who rule Triskelion from their equilateral triangular perch (c).<br />
In the 1990’s science fiction epic Babylon 5, the Triluminary was a crystal<br />
shaped as an equilateral triangle with a small chip at its center. The holiest<br />
of relics in Minbari society, it was a multifaceted device. Its primary function<br />
was to glow in the presence of the DNA of the Minbari prophet Valen. Its use<br />
brought the Earth-Minbari War to an end when it indicated that Commander<br />
Jeffrey Sinclair of Earth had Valen’s DNA and thus a Minbari soul thereby<br />
prompting a Minbari surrender. (Unlike Humans, Minbari do not kill Minbari!)<br />
A Triluminary (there were three of them in existence) was also used to induct<br />
a new member into the ruling Grey Council of Minbar. As shown in Figure<br />
1.49(a), Minbari Ambassador Delenn uses a Triluminary as part of a device<br />
which transforms her into a Human-Minbari hybrid in order to foster mutual<br />
understanding between the two races. The Triluminary is emblematic of the<br />
Minbari belief in the Trinity. As Zathras (caretaker of the Great Machine on<br />
Epsilon III) explains to Sinclair, Delenn and Captain John Sheridan: three<br />
castes, three languages, the Nine of the Grey Council (three times three), “All<br />
is three, as you are three, as you are one, as you are the One”. Sinclair then<br />
travels back in time one thousand years, uses the Triluminary (Figure 1.49(b))
28 History<br />
(a) (b)<br />
Figure 1.49: Triluminary (Babylon 5): (a) Delenn. (b) Sinclair.<br />
to become Valen (“a Minbari not born of Minbari”), and leads the forces of<br />
Light to victory over the Shadows.<br />
Figure 1.50: Symbol of the Deathly Hallows (Harry Potter)<br />
Figure 1.50 displays the symbol of the Deathly Hallows (composed of an<br />
equilateral triangle together with incircle and altitude) from the book/film<br />
Harry Potter and the Deathly Hallows which represents three magical items:<br />
the Elder Wand, the Resurrection Stone and the Cloak of Invisibility. He who<br />
unites the Hallows is thereby granted the power to elude Death.<br />
In retrospect, one is struck by the universal appeal of the equilateral triangle.<br />
Its appearances date back to the beginnings of recorded history and<br />
interest in it has transcended cultural boundaries. Since the equilateral triangle<br />
may be studied using only the rudiments of Mathematics, there is a certain<br />
temptation to dismiss it as mathematically trivial. However, there are aspects<br />
of the equilateral triangle that are both mathematically deep and stunningly<br />
beautiful. Let us now explore some of the many facets of this glistening jewel!
Chapter 2<br />
Mathematical Properties of the<br />
Equilateral Triangle<br />
Figure 2.1: Equilateral Triangle and Friends<br />
Property 1 (Basic Properties). Given an equilateral triangle of side s:<br />
perimeter (p) 3s<br />
altitude (a) s √ 3<br />
2<br />
area (A) s 2 √ 3<br />
4<br />
inradius (r) s √ 3<br />
6<br />
circumradius (R) s √ 3<br />
3<br />
2 π<br />
incircle area (Ar) s 12<br />
2 π<br />
circumcircle area (AR) s<br />
29<br />
3
30 Mathematical Properties<br />
The relation R = 2r, which is a consequence of the coincidence of the<br />
circumcenter (intersection of perpendicular bisectors) and the incenter (intersection<br />
of angle bisectors) with the centroid (intersection of medians which<br />
trisect one another), is but an outer manifestation of the hidden inner relations<br />
of Figure 2.1. It immediately implies that the area of the annular region<br />
between the circumcircle and the incircle is three times the area of the latter.<br />
In turn, this leads directly to the beautiful relation that the sum of the shaded<br />
areas equals the area of the incircle!<br />
(a) (b) (c)<br />
Figure 2.2: Construction of Equilateral Triangle: (a) Euclid [164]. (b) Hopkins<br />
[185]. (c) Weisstein [320].<br />
Property 2 (Construction of Equilateral Triangle). An equilateral may<br />
be constructed with straightedge and compass in at least three ways.<br />
1. Figure 2.2(a): Let AB be a given line segment. With center A and radius<br />
AB, construct circle BCD. With center B and radius BA, construct<br />
circle ACE. From their point of intersection C, draw line segments CA<br />
and CB. ∆ABC is equilateral [164].<br />
2. Figure 2.2(b): Given a circle with center F and radius FC, draw the arc<br />
DFE with center C and radius CF. With the same radius, and D and<br />
E as centers, set off points A and B. ∆ABC is equilateral [185].<br />
3. Figure 2.2(c): Draw a diameter OP0 of a circle and construct its perpendicular<br />
bisector P3OB; bisect OB in point D and extend the line P1P2<br />
through D and parallel to OP0. ∆P1P2P3 is then equilateral [320].<br />
Property 3 (Rusty Compass Construction). Some compasses are rusty,<br />
so that their opening cannot be changed. Given such a rusty compass whose<br />
opening is at least half the given side length AB (so that the constructed circles<br />
intersect), it is possible to extend the Euclidean construction to a corresponding<br />
five-circle construction [237].
Mathematical Properties 31<br />
A B<br />
D<br />
E F<br />
C<br />
Figure 2.3: Five-Circle Rusty Compass Construction<br />
All five circles in Figure 2.3 have the same radius. The first two are centered<br />
at the given points A and B. These two circles intersect at two points. Call<br />
one of them D and make it the center of a third circle which intersects the<br />
first two circles at two new points E and F (in addition to A and B) which<br />
again serve as centers of two additional circles. These last two circles intersect<br />
at D and one additional point, C. ∆ABC is equilateral. This follows since<br />
the central angle BFD is, by construction, equal to 60 ◦ , so that the inscribed<br />
angle BCD is equal to half of that, or 30 ◦ . The same is true of angle ACD,<br />
so that angle BCA is equal to 60 ◦ . By symmetry, triangle ABC is isosceles<br />
(AC = BC) thereby making ∆ABC equilateral. According to Pedoe [237, p.<br />
xxxvi], a student stumbled upon this construction while idly doodling in class,<br />
yet it has generated considerable research in rusty compass constructions.<br />
Figure 2.4: Equation of Equilateral Triangle [320]<br />
Property 4 (Equation of Equilateral Triangle). An equation for an equilateral<br />
triangle with R = 1 (s = √ 3) (Figure 2.4) [320]:<br />
max (−2y, y − x √ 3, y + x √ 3) = 1.
32 Mathematical Properties<br />
Figure 2.5: Trisection Through Bisection<br />
Property 5 (Trisection Through Bisection). In an equilateral triangle,<br />
the altitudes, angle bisectors, perpendicular bisectors and medians coincide.<br />
These three bisectors provide the only way to divide an equilateral triangle<br />
into two congruent pieces using a straight-line cut (Figure 2.5 left). Interestingly,<br />
these very same bisectors, suitably constrained, also provide the only two<br />
trisections of the equilateral triangle into congruent pieces using just straightline<br />
cuts (Figure 2.5 center and right).<br />
Figure 2.6: Triangular Numbers<br />
Property 6 (Triangular Numbers). The first six triangular numbers are<br />
on display in Figure 2.6. They are defined as:<br />
∆n =<br />
n�<br />
k =<br />
k=1<br />
n(n + 1)<br />
2<br />
=<br />
� �<br />
n + 1<br />
.<br />
2<br />
As such, ∆n solves the “handshake problem” of counting the number of<br />
handshakes in a room full of n + 1 people if each person shakes hands once<br />
with each other person. They are closely related to other figurate numbers<br />
[55]. For example, the sum of two consecutive triangular numbers is a square<br />
number with the sum being the square of their difference:<br />
∆n + ∆n−1 = 1 + 3 + 5 + · · · + (2n − 1) = n 2 = (∆n − ∆n−1) 2 .
Mathematical Properties 33<br />
Also, the sum of the first n triangular numbers is the nth tetrahedral number:<br />
n�<br />
k=1<br />
∆k =<br />
n(n + 1)(n + 2)<br />
.<br />
6<br />
The triangular numbers are also subject to many beautiful recurrence relations<br />
such as:<br />
∆ 2 n+∆ 2 n−1 = ∆ n 2, 3∆n+∆n−1 = ∆2n, 3∆n+∆n+1 = ∆2n+1, ∆ 2 n+1−∆ 2 n = (n+1) 3 .<br />
Finally, the triangular number ∆n = n + (n − 1) + · · · + 2 + 1 is the additive<br />
analog of the multiplicative factorial n! = n · (n − 1)···2 · 1 [320].<br />
(a) (b) (c)<br />
Figure 2.7: Equitriangular Unit of Area: (a) ETU. (b) Equilateral Area. (c)<br />
Scalene Area. [255]<br />
Property 7 (Equitriangular Unit of Area). Motivated by the identity<br />
∆n + ∆n−1 = n 2 , W. Roberts has introduced [255] the equitriangular unit of<br />
area (etu) shown in Figure 2.7(a).<br />
Naturally, the area of an equilateral triangle of side n is then equal to n 2<br />
etu, as seen in Figure 2.7(b). Referring to Figure 2.7(c), the area of a scalene<br />
triangle is simply ab etu where a is the 60 ◦ altitude of the vertex above base b.<br />
Property 8 (Viviani’s Theorem). For any point inside an equilateral triangle,<br />
the sum of the distances to the sides is equal to the height of the triangle.<br />
(Figure 2.8: PE + PF + PG = h.)<br />
The same is true if the point lies outside the triangle so long as signed<br />
distances are employed [322]. Conversely, if inside a triangle there is a circular<br />
region in which the sum of the distances from any point to the sides of the<br />
triangle is constant then the triangle is equilateral [48].<br />
Property 9 (Ternary Diagram). Viviani’s Theorem implies that lines parallel<br />
to the sides of an equilateral triangle provide (homogeneous/barycentric/<br />
areal/trilinear) coordinates [275] for ternary diagrams for representing three<br />
quantities A, B, C whose sum is a constant (which can be normalized to unity).
34 Mathematical Properties<br />
Figure 2.8: Viviani’s Theorem Figure 2.9: Ternary Diagram<br />
With reference to Figure 2.9, each variable is assigned to a vertex as well<br />
as to the clockwise-adjacent edge. These diagrams are employed in petrology,<br />
mineralogy, metallurgy and other physical sciences to portray the composition<br />
of systems composed of three species. In population genetics, it is called the<br />
de Finetti diagram and, in game theory, it is called the simplex plot. Ample<br />
specific instances will be provided in Chapter 3: Applications.<br />
Figure 2.10: Ellipsoidal Shape [243]<br />
Property 10 (Ellipsoidal Shape). The shape of an ellipsoid with semi-axes<br />
a, b and c depends only on the ratios a : b : c. In order to visualize the variety<br />
of these shapes, we may consider a, b and c as homogeneous coordinates ([256,<br />
pp. 16-20], [257, pp. 179-190]) of a point (a, b, c) in a plane and consider<br />
only that part of this plane where a ≥ b ≥ c ≥ 0. These inequalities delimit a<br />
triangle which can be made equilateral by a suitable choice of coordinate system<br />
(Figure 2.10) [243, p. 37].
Mathematical Properties 35<br />
The points of this triangle are in one-to-one correspondence with the different<br />
ellipsoidal shapes. The interior points correspond to non-degenerate<br />
ellipsoids with three different axes, the boundary points to ellipsoids of revolution<br />
or to degenerate ellipsoids. The three vertices represent the sphere, the<br />
circular disk and the “needle” (segment of a straight line), respectively.<br />
Figure 2.11: Morley’s Theorem<br />
Property 11 (Morley’s Theorem). The adjacent pairs of the trisectors of<br />
the interior angles of a triangle always meet at the vertices of an equilateral<br />
triangle [178] called the first Morley triangle (Figure 2.11).<br />
If, instead, the exterior angle trisectors are used then another equilateral<br />
triangle is formed (the second Morley triangle) and, moreover, the intersections<br />
of the sides of this triangle with the external trisectors form three additional<br />
equilateral triangles [322].<br />
Property 12 (Fermat-Torricelli Problem). In 1629, Fermat challenged<br />
Torricelli to find a point whose total sum of distances from the vertices of a<br />
triangle is a minimum [224]. I.e., determine a point X (Fermat point) in a<br />
given triangle ABC such that the sum XA + XB + XC is a minimum [178].<br />
If the angle at one vertex is greater than or equal to 120 ◦ then the Fermat<br />
point coincides with this vertex. Otherwise, the Fermat point coincides with<br />
the so-called (inner) isogonic center (X in Figure 2.12) which may be found by<br />
constructing outward pointing equilateral triangles on the sides of ∆ABC and
36 Mathematical Properties<br />
Figure 2.12: Fermat Point<br />
connecting each vertex of the original triangle to the new vertex of the opposite<br />
equilateral triangle. These three segments are of equal length and intersect at<br />
the isogonic center where they are inclined at 60 ◦ to one another. At the<br />
isogonic center, each side of the original triangle subtends an angle of 120 ◦ .<br />
Also, the isogonic center lies at the common intersection of the circumcircles of<br />
the three equilateral triangles [322]. The algebraic sum of the distances from<br />
the isogonic center to the vertices of the triangle equals the length of the equal<br />
segments from the latter to the opposite vertices of the equilateral triangles<br />
[189].<br />
Property 13 (Largest Circumscribed Equilateral Triangle). If we connect<br />
the isogonic center of an arbitrary triangle with its vertices and draw lines<br />
through the latter perpendicular to the connectors then these lines intersect to<br />
form the largest equilateral triangle circumscribing the given triangle [189].<br />
This is the antipedal triangle of the isogonic center with respect to the given<br />
triangle.<br />
Property 14 (Napoleon’s Theorem). On each side of an arbitrary triangle,<br />
construct an equilateral triangle pointing outwards. The centers of these three<br />
triangles form an equilateral triangle [178] called the outer Napoleon triangle<br />
(Figure 2.13(a)).
Mathematical Properties 37<br />
(a) (b) (c)<br />
Figure 2.13: Napoleon’s Theorem [322, 178, 245]<br />
If, instead, the three equilateral triangles point inwards then another equilateral<br />
triangle is formed (the inner Napoleon triangle) and, moreover, the two<br />
Napoleon triangles share the same center with the original equilateral triangle<br />
and the difference in their areas is equal to the area of the original triangle<br />
[322]. (See [71] for a most interesting conjectured provenance for this theorem.)<br />
Also, the lines joining a vertex of either Napoleon triangle with the remote vertex<br />
of the original triangle are concurrent (Figure 2.13(b)) [245]. Finally, the<br />
lines joining each vertex of either Napoleon triangle to the new vertex of the<br />
corresponding equilateral triangle drawn on each side of the original equilateral<br />
triangle are conccurrent and, moreover, the point of concurrency is the<br />
circumcenter of the original equilateral triangle (Figure 2.13(c)) [245].<br />
(a)<br />
(b)<br />
Figure 2.14: Parallelogram Properties [245].<br />
Property 15 (Parallelogram Properties). With reference to Figure 2.14(a),<br />
equilateral triangles BCE and CDF are constructed on sides BC and CD, respectively,<br />
of a parallelogram ABCD. Since triangles BEA, CEF and DAF<br />
are congruent, triangle AEF is equilateral [245]. With reference to Figure
38 Mathematical Properties<br />
2.14(b) left/right, construct equilateral triangles pointing outwards/inwards on<br />
the sides of an oriented parallelogram ABCD giving parallelogram XY ZW.<br />
Then, if inward/outward pointing equilateral triangles are drawn on the sides<br />
of oriented parallelogram XY ZW, the resulting parallelogram is just ABCD<br />
again [182].<br />
(a)<br />
Figure 2.15: (a) Isodynamic (Apollonius) Points. (b) Pedal Triangle of First<br />
Isodynamic Point<br />
Property 16 (Pedal Triangles of Isodynamic Points). With reference<br />
to ∆ABC of Figure 2.15(a), let U and V be the points on BC met by the<br />
interior and exterior bisectors of ∠A. The circle having diameter UV is called<br />
the A-Apollonian circle [6]. The B- and C-Apollonian circles are likewise<br />
defined. These three Apollonian circles intersect at the first (J) and second<br />
(J ′ ) isodynamic (Apollonius) points [116]. With reference to Figure 2.15(b),<br />
connecting the feet of the perpendiculars from the first isodynamic point, I ′ ,<br />
to the sides of ∆ABC produces its pedal triangle which is always equilateral<br />
[189]. The same is true for the pedal triangle of the second isodynamic point.<br />
This theorem generalizes as follows: The pedal triangle of any of the four<br />
points A, B, C, I ′ with respect to the triangle formed by the remaining points is<br />
equilateral [46, p. 303].<br />
Property 17 (The Machine for Questions and Answers). In 2006, D.<br />
Dekov created a computer program, The Machine for Questions and Answers,<br />
and used it to produce The Computer-Generated Encyclopedia of Euclidean<br />
Geometry which contains the following results pertinent to equilateral triangles<br />
[73]. (Note: Connecting a point to the three vertices of a given triangle creates<br />
three new triangulation triangles associated with this point. Also, deleting the<br />
cevian triangle [6, p. 160] of a point with respect to a given triangle leaves<br />
three new corner triangles associated with this point.)<br />
(b)
Mathematical Properties 39<br />
• The pedal triangle of the first/second isodynamic point is equilateral.<br />
• The antipedal triangle of the inner/outer Fermat point is equilateral.<br />
• The circumcevian triangle of the first/second isodynamic point is equilateral.<br />
• The inner/outer Napoleon triangle is equilateral.<br />
• The triangle formed by the circumcenters of the triangulation triangles<br />
of the inner/outer Fermat point is equilateral.<br />
• The triangle formed by the nine-point centers of the triangulation triangles<br />
of the inner/outer Fermat point is equilateral.<br />
• The triangle formed by the first/second isodynamic points of the corner<br />
triangles of the orthocenter is equilateral.<br />
• The triangle of the inner/outer Fermat points of the corner triangles of<br />
the Gergonne point is equilateral.<br />
• The triangle formed by the first/second isodynamic points of the anticevian<br />
corner triangles of the incenter is equilateral.<br />
• The triangle formed by the inner Fermat points of the anticevian corner<br />
triangles of the second isodynamic point is equilateral.<br />
• The triangle formed by the reflections of the first isodynamic point in<br />
the sides of a given triangle is equilateral.<br />
Property 18 (Musselman’s Theorem). In May 1932, J. R. Musselman<br />
published a collection of results pertaining to the generation of equilateral triangles<br />
from equilateral triangles [223]. In the results to follow, a positively/negatively<br />
equilateral triangle P1P2P3 is one whose vertices rotate into each other in a<br />
counterclockwise/clockwise direction, respectively.<br />
• Connected with two given positively equilateral triangles of any size or<br />
position, we can find three other equilateral triangles. Specifically, if<br />
A1A2A3 and B1B2B3 are the vertices of the two given triangles then the<br />
midpoints of {A1B1, A2B2, A3B3} are the vertices of an equilateral triangle<br />
as are the midpoints of {A1B2, A2B3, A3B1} and {A1B3, A2B1, A3B2}.<br />
• Connected with three given positively equilateral triangles of any size<br />
or position with vertices A1A2A3, B1B2B3 and C1C2C3, we can find 18<br />
other equilateral triangles. This we do as follows.
40 Mathematical Properties<br />
– Combining by pairs in all possible, three, ways and connecting midpoints<br />
as above, we obtain 9 equilateral triangles.<br />
– Also, the centroids of the three triangles A1B1C1, A2B2C2, A3B3C3<br />
themselves form an equilateral triangle. Moreover, the same is true<br />
if we fix the Ai and successively permute the Bi and Ci, for a total<br />
of 9 ways in which the Ai, Bi, Ci can be arranged so that the<br />
centroids of the three resulting triangles form an equilateral triangle.<br />
Incidentally, the centroid of each of the nine equilateral triangles<br />
thus formed is the same point!<br />
• If the above three positively equilateral triangles be now so placed in the<br />
plane that A1B1C1 forms a positively equilateral triangle, then there exist<br />
fifty-five equilateral triangles associated with this configuration! This we<br />
establish as follows.<br />
– First of all, the centroids of the three triangles B1C1A1, B2C2B1,<br />
B3C3C1 form a positively equilateral triangle. To obtain the 9 equilateral<br />
triangles that can be thus formed, keep the last letters, A1,<br />
B1, C1, fixed and cyclically permute the remaining Bi and Ci. (E.g.,<br />
the centroid of B1C2B1 is the point on the segment B1C2 one third<br />
of the distance B1C2 from B1.)<br />
– Secondly, the midpoints of B3C2, A2C3, A3B2 form an additional<br />
equilateral triangle.<br />
– Finally, the existence of fifty-five equilateral triangles connected<br />
with the figure of three equilateral triangles, so placed in a plane<br />
that one vertex of each also forms an equilateral triangle, is now<br />
apparent. Taking the four given equilateral triangles in (six) pairs,<br />
each pair produces 3 additional equilateral triangles, or 18 in all. If<br />
we take the equilateral triangles in (four) triples, each triple yields<br />
9 additional equilateral triangles, or 36 in all. Finally, there is the<br />
additional equilateral triangle just noted, thus making a total of 55<br />
equilateral triangles which be easily constructed.<br />
• Given two positively equilateral triangles, A1B1C1 and A2B2C2, of any<br />
size or position in the plane, if we construct the positively/negatively<br />
equilateral triangles A1A2A3, B1B2B3 and C1C2C3 then A3B3C3 is itself<br />
a positively equilateral triangle.<br />
Property 19 (Distances from Vertices). The symmetric equation<br />
3(a 4 + b 4 + c 4 + d 4 ) = (a 2 + b 2 + c 2 + d 2 ) 2<br />
relates the side of an equilateral triangle to the distances of a point from its<br />
three corners [129, p. 65]. Any three variables can be taken for the three<br />
distances and solving for the fourth then gives the triangle’s side.
Mathematical Properties 41<br />
The simplest solution in integers is 3, 5, 7, 8. One of a, b, c, d is divisible<br />
by 3, one by 5, one by 7 and one by 8 [159, p. 183], although they need not<br />
be distinct: (57,65,73,112) & (73,88,95,147).<br />
Figure 2.16: Largest Inscribed Square [138]<br />
Property 20 (Largest Inscribed Square). Figure 2.16 displays the largest<br />
square that can be inscribed in an equilateral triangle [138]. Its side is of length<br />
2 √ 3 − 3 ≈ 0.4641 and its area is 21 − 12 √ 3 ≈ 0.2154.<br />
This is slightly smaller than the largest inscribed rectangle which can be<br />
constructed by dropping perpendiculars from the midpoints of any two sides<br />
to the third side. The feet of these perpendiculars together with the original<br />
two midpoints form the vertices of the largest inscribed rectangle whose base is<br />
half the triangle’s base and whose area is half the triangle’s area, √ 3<br />
8<br />
≈ 0.2165.<br />
Moreover, this is the maximum area rectangle that fits inside the triangle<br />
regardless of whether or not it is inscribed.<br />
Property 21 (Triangle in Square). The smallest equilateral triangle inscribable<br />
in a unit square (Figure 2.17 left) has sides equal to unity and area<br />
equal to √ 3 ≈ 0.4330. The largest such equilateral triangle (Figure 2.17 right)<br />
4<br />
has sides of length √ 6 − √ 2 ≈ 1.0353 and area equal to 2 √ 3 − 3 ≈ 0.4641.<br />
[214]<br />
The more general problem of fitting the largest equilateral triangle into a<br />
given rectangle has been solved by Wetzel [326].
42 Mathematical Properties<br />
Figure 2.17: Inscribed Triangles: Smallest (Left) and Largest (Right) [214]<br />
(a)<br />
Figure 2.18: Syzygies: (a) Length Equals Area. (b) Related Constructions.<br />
[138]<br />
(b)
Mathematical Properties 43<br />
Property 22 (Syzygies). (a) As can be concluded from the previous two<br />
Properties, M. Gardner has observed [138] that length of the side of the largest<br />
square that fits into an equilateral triangle of side 1 is the same as the area<br />
of the largest equilateral triangle that fits inside a unit square, in both cases<br />
2 √ 3−3. Subsequently, J. Conway gave the dissection proof illustrated in Figure<br />
2.18(a). The area of the shaded parallelogram is equal to 2 √ 3 − 3, the length<br />
of the side of the inscribed square, and may be dissected into three pieces as<br />
shown which fit precisely into the inscribed equilateral triangle. (b) B. Cipra<br />
discovered [138] that the endpoints of the baseline which constructs the largest<br />
square in the equilateral triangle also mark the points on the top of the larger<br />
square that are the top corners of the two maximum equilateral triangles that fit<br />
within the unit square (Figure 2.18(b)). Thus, juxtaposing these two diagrams<br />
produces a bilaterally symmetrical pattern illustrating the intimate connections<br />
between these two constructions.<br />
Figure 2.19: Equilateral Triangles and Triangles: (a)-(c) Three Possible Configurations.<br />
(d) Equal Equilateral Triangles. [188]<br />
Property 23 (Equilateral Triangles and Triangles). In 1964, H. Steinhaus<br />
[292] posed the problem of finding a necessary and sufficient condition on<br />
the six sides a, b, c, a ′ , b ′ , c ′ for triangle T ′ with sides a ′ , b ′ , c ′ to fit in triangle T<br />
with sides a, b, c. In 1993, K. Post [247] succeeded by providing 18 inequalities<br />
whose disjunction is both necessary and sufficient. Post’s proof hinges on the<br />
theorem that if one triangle fits within a second in any way whatsoever, then it<br />
also fits is such a way that one of its sides lies on a side of the containing triangle.<br />
Jerrard and Wetzel [188] have given geometric conditions for the more<br />
specialized problems of how large an equilateral triangle can fit into a given<br />
arbitrary triangle and how small an equilateral triangle can contain a given<br />
arbitrary triangle.<br />
By Post’s Theorem, the largest equilateral triangle ∆in that fits in the<br />
given equilateral triangle T = ABC does so with one side along a side of T,<br />
and T fits in the smallest equilateral triangle ∆out that contains it with one<br />
of its sides along a side of ∆out. Defining sin to be the side of ∆in and s z in to<br />
(d)
44 Mathematical Properties<br />
be the side of the largest equilateral triangle ∆ z in that fits in T with one side<br />
along side z ∈ {a, b, c} of T, we have that sin = max{s a in, s b in, s c in}. Likewise,<br />
defining sout to be the side of ∆out and s z out to be the side of the smallest<br />
equilateral triangle ∆ z in containing T with one side along side z ∈ {a, b, c} of<br />
T, we have that sout = min{s a out, s b out, s c out}. They establish the surprising fact<br />
that ∆in and ∆out always rest on the same side of T. Curiously, the area of<br />
T is the geometric mean of the areas of ∆ z in and ∆ z out for each side z of T<br />
and thus of the areas of ∆in and ∆out. Figures 2.19(a-c) display the three<br />
possible configurations where z is the longest, shortest and median side of T,<br />
respectively.<br />
They also show that ∆in and ∆out always lie on either the longest or the<br />
shortest side of T. If the median angle of T is at most 60 ◦ then they rest<br />
on the longest side. Otherwise, there is a complicated condition involving the<br />
median angle and its adjacent sides which determines whether they rest on the<br />
longest or the shortest side of T. Along the way, they work out the ordering<br />
relations among s a in, s b in, s c in and s a out, s b out, s c out which then permits the explicit<br />
calculation of sin and sout. It follows that if an equilateral triangle ∆ of side s<br />
and a triangle T are given, then ∆ fits within T precisely when s ≤ sin and T<br />
fits within ∆ precisely when s ≥ sout. Finally, they establish that the isosceles<br />
triangle with apex angle 20 ◦ is the unique nonequilateral triangle for which the<br />
three inner/outer maximal equilateral triangles are congruent (Figure 2.19(d)).<br />
(a)<br />
(b) (c) (d) (e)<br />
Figure 2.20: Packings: (a) Squares in Triangle. (b) Triangles in Square. (c)<br />
Circles in Triangle. (d) Triangles in Circle. (e) Triangles in Triangle. [110]<br />
Property 24 (Packings). The packing of congruent equilateral triangles,<br />
squares and circles into an equilateral triangle, a square or a circle has received<br />
considerable attention [110]. A sampling of some of the densest known<br />
packings is offered by Figure 2.20: (a) 3 squares in an equilateral triangle<br />
(Friedman 1997), (b) 3 equilateral triangles in a square (Friedman 1996), (c)<br />
8 circles in an equilateral triangle (Melissen 1993), (d) 8 equilateral triangles<br />
in a circle (Morandi 2008), (e) 5 equilateral triangles in an equilateral triangle<br />
(Friedman 1997) [110]. The most exhaustively studied case is that of packing<br />
congruent circles into an equilateral triangle [220, 150].
Mathematical Properties 45<br />
(a)<br />
(b)<br />
(c)<br />
(d) (e)<br />
Figure 2.21: Coverings: (a) Squares on Triangle. (b) Triangles on Square. (c)<br />
Circles on Triangle. (d) Triangles on Circle. (e) Triangles on Triangle. [110]<br />
Property 25 (Coverings). The covering of an equilateral triangle, a square<br />
or a circle by congruent equilateral triangles, squares and circles has also received<br />
a fair amount of attention [110]. A sampling of some of the known<br />
optimal coverings is offered by Figure 2.21: (a) 3 squares on an equilateral triangle<br />
(Cantrell 2002), (b) 4 equilateral triangles on a square (Friedman 2002),<br />
(c) 4 circles on an equilateral triangle (Melissen 1997), (d) 4 equilateral triangles<br />
on a circle (Green 1999), (e) 7 equilateral triangles on an equilateral<br />
triangle (Friedman 1999) [110].<br />
Property 26 (Covering Properties). A convex region that contains a congruent<br />
copy of each curve of a specified family is called a cover for the family.<br />
The following results concern equilateral triangular covers.<br />
• Every plane set of diameter one can be completely covered with an equilateral<br />
triangle of side √ 3 ≈ 1.7321 [178].<br />
• The smallest equilateral triangle that can cover every triangle of diameter<br />
one has side (2 cos 10 ◦ )/ √ 3 ≈ 1.1372 [325].<br />
• The smallest triangular cover for the family of all closed curves of length<br />
two is the equilateral triangle of side 2 √ 3/π ≈ 1.1027 [114], a result that<br />
follows from an inequality published in 1957 by Eggleston [94, p. 157].<br />
• The smallest equilateral triangle that can cover every triangle of perimeter<br />
two has side 2/m0 ≈ 1.002851, where m0 is the global minimum of<br />
the the trigonometric function √ 3·(1+sin x)·sec<br />
(π − x) on the interval<br />
2 6<br />
[0, π/6] [325].<br />
Property 27 (Blaschke’s Theorem). The width of a closed convex curve<br />
in a given direction is the distance between the two closest parallel lines, perpendicular<br />
to that direction, which enclose the curve. Blaschke proved that any<br />
closed convex curve whose minimum width is 1 unit or more contains a circle<br />
of diameter 2/3 unit. An equilateral triangle contains just such a circle (its<br />
incircle), so the limit of 2/3 is the best possible [322].
46 Mathematical Properties<br />
Figure 2.22: Euler Line Cuts Off Equilateral Triangle<br />
Property 28 (Euler Line Cuts Off Equilateral Triangle). If a triangle<br />
is not equilateral then the orthocenter (intersection of altitudes), the centroid<br />
(intersection of medians) and the circumcenter (intersection of perpendicular<br />
bisectors) are collinear [58]. (For an equilateral triangle, these three points<br />
coincide.) Many other important points associated with a triangle, such as the<br />
nine-point center, also lie on this so-called Euler line [6]. In a triangle with a<br />
60 ◦ angle, the Euler line cuts off an equilateral triangle [28] (see Figure 2.22).<br />
Property 29 (Incircle-Triangle Iteration). Let ∆A0B0C0 be arbitrary. Let<br />
the points of contact with its incircle be A1, B1, C1. Let the points of contact of<br />
∆A1B1C1 with its incircle be A2, B2, C2, and so on. This sequence of triangles<br />
shrinks by a factor of 1/2 at each iteration and approaches equiangularity in<br />
the limit [47].<br />
Property 30 (Excentral Triangle Iteration). The bisector of any interior<br />
angle of a triangle and those of the exterior angles at the other two vertices<br />
are concurrent at a point outside the triangle. These three points are called<br />
excenters and they are the vertices of the excentral triangle. Commencing<br />
with an arbitrary triangle, construct its excentral triangle, then construct the<br />
excentral triangle of this excentral triangle, and so on. These excentral triangles<br />
approach an equilateral triangle [189].
Mathematical Properties 47<br />
Figure 2.23: Abutting Equilateral Triangles [179]<br />
Property 31 (Abutting Equilateral Triangles). Equilateral triangles of<br />
sides 1, 3, 5, . . . , 2n − 1, . . . are placed end-to-end along a straight line<br />
(Figure 2.23). The vertices which do not lie on the line all lie on a parabola<br />
and their focal radii are all integers [179].<br />
Figure 2.24: Circumscribing Rectangle [181]<br />
Property 32 (Circumscribing Rectangle). Around any equilateral triangle<br />
ABC, circumscribe a rectangle PBQR (Figure 2.24). In general, each side of<br />
ABC cuts off a right triangle from the rectangle. The areas of the two smaller<br />
right triangles always add up to the area of the largest one (X = Y +Z) [181].
48 Mathematical Properties<br />
Figure 2.25: Equilic Quadrilateral [181]<br />
Property 33 (Equilic Quadrilaterals). With reference to Figure 2.25, a<br />
quadrilateral ABCD is equilic if AD = BC and ∠A + ∠B = 120 ◦ . Figure<br />
2.26(a): The midpoints P, Q and R of the diagonals and the side CD always<br />
determine an equilateral triangle. Figure 2.26(b): If an equilateral triangle<br />
PCD is drawn outwardly on CD then ∆PAB is also equilateral [181].<br />
(a)<br />
Figure 2.26: (a) Equilic Midpoints. (b) Equilic Triangles. [181]<br />
Property 34 (The Only Rational Triangle). If a triangle has side lengths<br />
which are all rational numbers and angles which are all a rational number of<br />
degrees then the triangle must be equilateral [55]!<br />
Property 35 (Six Triangles). From an arbitrary point in an equilateral<br />
triangle, segments to the vertices and perpendiculars to the sides partition the<br />
triangle into six smaller triangles A, B, C, D, E, F (see Figure 2.27 left).<br />
Claim [183]:<br />
A + C + E = B + D + F.<br />
(b)
Mathematical Properties 49<br />
Figure 2.27: Six Triangles [183]<br />
Drawing three additional lines through the selected point which are parallel<br />
to the sides of the original triangle partitions it into three parallelograms and<br />
three small equilateral triangles (Figure 2.27 right). Since the areas of the<br />
parallelograms are bisected by their diagonals and the equilateral triangles by<br />
their altitudes,<br />
A + C + E = x + a + y + b + z + c = B + D + F.<br />
Figure 2.28: Pompeiu’s Theorem [184]<br />
Property 36 (Pompeiu’s Theorem). If P is an arbitrary point in an equilateral<br />
triangle ABC then there exists a triangle with sides of length PA, PB,<br />
PC [184].<br />
Draw segments PL, PM, PN parallel to the sides of the triangle (Figure<br />
2.28). Then, the trapezoids PMAN, PNBL, PLCM are isosceles and thus<br />
have equal diagonals. Hence, PA = MN, PB=LN, PC = LM and ∆LMN<br />
is the required triangle. Note that the theorem remains valid for any point P<br />
in the plane of ∆ABC [184] and that the triangle is degenerate if and only if<br />
P lies on the circumcircle of ∆ABC [267].
50 Mathematical Properties<br />
Figure 2.29: Random Point [177]<br />
Property 37 (Random Point). A point P is chosen at random inside an<br />
equilateral triangle. Perpendiculars from P to the sides of the triangle meet<br />
these sides at points X, Y , Z. The probability that a triangle with sides PX,<br />
PY , PZ exists is equal to 1<br />
4 [177].<br />
As shown in Figure 2.29, the segments satisfy the triangle inequality if and<br />
only if the point lies in the shaded region whose area is one fourth that of the<br />
original triangle [122]. Compare this result to Pompeiu’s Theorem!<br />
Property 38 (Gauss Plane). In the Gauss (complex) plane [81], ∆ABC is<br />
equilateral if and only if<br />
(b − a)λ 2 ± = (c − b)λ± = a − c; λ± := (−1 ± ı √ 3)/2.<br />
For λ±, ∆ABC is described counterclockwise/clockwise, respectively.<br />
Property 39 (Gauss’ Theorem on Triangular Numbers). In his diary<br />
of July 10, 1796, Gauss wrote [290]:<br />
“EΥPHKA! num = ∆ + ∆ + ∆.”<br />
I.e., “Eureka! Every positive integer is the sum of at most three triangular<br />
numbers.”<br />
As early as 1638, Fermat conjectured much more in his polygonal number<br />
theorem [319]: “Every positive integer is a sum of at most three triangular<br />
numbers, four square numbers, five pentagonal numbers, and n n-polygonal<br />
numbers.” (Alas, his margin was once again too narrow to hold his proof!)<br />
Jacobi and Lagrange proved the square case in 1772, Gauss the triangular case<br />
in 1796, and Cauchy the general case in 1813 [320].
Mathematical Properties 51<br />
Figure 2.30: Equilateral Shadows [180]<br />
Property 40 (Equilateral Shadows). Any triangle can be orthogonally projected<br />
onto an equilateral triangle [180]. Moreover, under the inverse of this<br />
transformation, the incircle of the equilateral triangle is mapped to the “midpoint<br />
ellipse” of the original triangle with center at the triangle centroid and<br />
tangent to the triangle sides at their midpoints (Figure 2.30).<br />
Note that this demonstrates that if we cut a triangle from a piece of paper<br />
and hold it under the noonday sun then we can always position the triangle<br />
so that its shadow is an equilateral triangle.<br />
Figure 2.31: Fundamental Theorem of Affine Geometry [33]<br />
Property 41 (Affine Geometry). All triangles are affine-congruent [33]. In<br />
particular, any triangle may be affinely mapped onto any equilateral triangle<br />
(Figure 2.31).<br />
This theorem is of fundamental importance in the theory of Riemann surfaces.<br />
E.g. [288, p. 113]:<br />
Theorem 2.1 (Riemann Surfaces). If an arbitrary manifold M is given<br />
which is both triangulable and orientable then it is possible to define an analytic<br />
structure on M which makes it into a Riemann surface.
52 Mathematical Properties<br />
Figure 2.32: Largest Inscribed Triangle and Least-Diameter Decomposition of<br />
the Open Disk [4]<br />
Property 42 (Largest Inscribed Triangle). The triangle of largest area<br />
that is inscribed in a given circle is the equilateral triangle (Figure 2.32) [249].<br />
Property 43 (Planar Soap Bubble Clusters). An inscribed equilateral<br />
triangle (Figure 2.32) provides a least-diameter smooth decomposition of the<br />
open unit disk into relatively closed sets that meet at most two at a point [4].<br />
Property 44 (Jung’s Theorem). Let d be the (finite) diameter of a planar<br />
set and let r be the radius of its smallest enclosing circle. Then, [249]<br />
r ≤ d √ 3 .<br />
Since the circumcircle is the smallest enclosing circle for an equilateral triangle<br />
(Figure 2.32), this bound cannot be diminished.<br />
Property 45 (Isoperimetric Theorem for Triangles). Among triangles<br />
of a given perimeter, the equilateral triangle has the largest area [191]. Equivalently,<br />
among all triangles of a given area, the equilateral triangle has the<br />
shortest perimeter [228].<br />
Property 46 (A Triangle Inequality). If A is the area and L the perimeter<br />
of a triangle then<br />
A ≤ √ 3L 2 /36,<br />
with equality if and only if the triangle is equilateral [228].<br />
Property 47 (Euler’s Inequality). If r and R are the radii of the inscribed<br />
and circumscribed circles of a triangle then<br />
R ≥ 2r,<br />
with equality if and only if the triangle is equilateral [191, 228].
Mathematical Properties 53<br />
Property 48 (Erdös-Mordell Inequality). Let R1, R2, R3 be the distances<br />
to the three vertices of a triangle from any interior point P. Let r1, r2, r3 be<br />
the distances from P to the three sides. Then<br />
R1 + R2 + R3 ≥ 2(r1 + r2 + r3),<br />
with equality if and only if the triangle is equilateral and P is its centroid<br />
[191, 228].<br />
Property 49 (Blundon’s Inequality). In any triangle ABC with circumradius<br />
R, inradius r and semi-perimeter σ, we have that<br />
σ ≤ 2R + (3 √ 3 − 4)r,<br />
with equality if and only if ABC is equilateral [26].<br />
Property 50 (Garfunkel-Bankoff Inequality). If Ai (i = 1, 2, 3) are the<br />
angles of an arbitrary triangle, then we have<br />
3�<br />
i=1<br />
2 Ai<br />
tan<br />
2<br />
≥ 2 − 8<br />
3�<br />
i=1<br />
sin Ai<br />
2 ,<br />
with equality if and only if ABC is equilateral [331].<br />
Property 51 (Improved Leunberger Inequality). If si (i = 1, 2, 3) are<br />
the sides of an arbitrary triangle with circumradius R and inradius r, then we<br />
have<br />
3�<br />
√<br />
1 25Rr − 2r2 ≥ ,<br />
4Rr<br />
i=1<br />
with equality if and only if ABC is equilateral [331].<br />
si<br />
Figure 2.33: Shortest Bisecting Path [161]<br />
Property 52 (Shortest Bisecting Path). The shortest path across an equilateral<br />
triangle of side s which bisects its area is given by a circular arc with<br />
center at a vertex and with radius chosen to bisect the area (Figure 2.33) [228].
54 Mathematical Properties<br />
Figure 2.34: Smallest Inscribed Triangle [57, 62]<br />
This radius is equal to s ·<br />
� 3 √ 3<br />
4π<br />
[161] so that the circular arc has length<br />
.673...s which is much shorter than either the .707...s length of the parallel<br />
bisector or the .866...s length of the altitude.<br />
Property 53 (Smallest Inscribed Triangle). The problem of finding the<br />
triangle of minimum perimeter inscribed in a given acute triangle [62] was<br />
posed by Giulio Fagnano and solved using calculus by his son Giovanni Fagnano<br />
in 1775 [224]. (An inscribed triangle being one with a vertex on each side of<br />
the given triangle.) The solution is given by the orthic/pedal triangle of the<br />
given acute triangle (Figure 2.34 left).<br />
Later, H. A. Schwarz provided a geometric proof using mirror reflections<br />
[57]. Call the process illustrated on the left of Figure 2.34 the pedal mapping.<br />
Then, the unique fixed point of the pedal mapping is the equilateral triangle<br />
[194]. That is, the equilateral triangle is the only triangle that maintains its<br />
form under the pedal mapping. Also, the equilateral triangle is the only triangle<br />
for which successive pedal iterates are all acute [205]. Finally, the maximal<br />
ratio of the perimeter of the pedal triangle to the perimeter of the given acute<br />
triangle is 1/2 and the unique maximizer is given by the equilateral triangle<br />
[158]. For a given equilateral triangle, the orthic/pedal triangle coincides with<br />
the medial triangle which is itself equilateral (Figure 2.34 right).<br />
Property 54 (Closed Light Paths [57]). The walls of an equilateral triangular<br />
room are mirrored. If a light beam emanates from the midpoint of a<br />
wall at an angle of 60 ◦ , it is reflected twice and returns to its point of origin<br />
by following a path along the pedal triangle (see Figure 2.34 right) of the room.<br />
If it originates from any other point along the boundary (exclusive of corners)<br />
at an angle of 60 ◦ , it is reflected five times and returns to its point of origin<br />
by following a path everywhere parallel to a wall (Figure 2.35) [57].
Mathematical Properties 55<br />
Figure 2.35: Closed Light Paths [57]<br />
Figure 2.36: Erdös-Moser Configuration [235]<br />
Property 55 (Erdös-Moser Configuration). An equilateral triangle of<br />
side-length one is called a unit triangle. A set of points S is said to span<br />
a unit triangle T if the vertices of T belong to S. n points in the plane are said<br />
to be in strictly convex position if they form the vertex set of a convex polygon<br />
for which each of the points is a corner. Pach and Pinchasi [235] have proved<br />
that any set of n points in strictly convex position in the plane has at most<br />
⌊2(n−1)/3⌋ triples that span unit triangles. Moreover, this bound is sharp for<br />
each n > 0.<br />
This maximum is attained by the Erdös-Moser configuration of Figure 2.36.<br />
This configuration contains ⌊(n − 1)/3⌋ congruent copies of a rhombus with<br />
side-length one and obtuse angle 2π/3, rotated by small angles around one of<br />
its vertices belonging to such an angle [235].<br />
Property 56 (Reuleaux Triangle). With reference to Figure 2.37(a), the<br />
Reuleaux triangle is obtained by replacing each side of an equilateral triangle by<br />
a circular arc with center at the opposite vertex and radius equal to the length<br />
of the side [125].
56 Mathematical Properties<br />
(a) (b)<br />
Figure 2.37: Curves of Constant Width: (a) Sharp Reuleaux Triangle. (b)<br />
Rounded Reuleaux Triangle.<br />
Like the circle, it is a curve of constant breadth, the breadth (width) being<br />
equal to the length of the triangle side [291]. Whereas the circle encloses<br />
the largest area amongst constant-width curves with a fixed width, w, the<br />
Reuleaux triangle encloses the smallest area amongst such curves (Blaschke-<br />
Lebesgue Theorem) [39]. (The enclosed area is equal to (π− √ 3)w 2 /2 [125]; by<br />
Barbier’s theorem [39], the perimeter equals πw) Moreover, a constant-width<br />
curve cannot be more pointed than 120 ◦ , with the Reuleaux triangle being the<br />
only one with a corner of 120 ◦ [249]. Like all curves of constant-width, the<br />
Reuleaux triangle is a rotor for a square, i.e. it can be rotated so as to maintain<br />
contact with the sides of the square, but the center of rotation is not fixed [241].<br />
In the case of the Reuleaux triangle, the square with rounded corners that is<br />
out swept out has area that is approximately equal to .9877 times the area of<br />
the square. (See Application 25.) The corners of the Reuleaux triangle can<br />
be smoothed by extending each side of the equilateral triangle a fixed distance<br />
at each end and then constructing six circular arcs centered at the vertices<br />
as shown in Figure 2.37(b). The constant-width of the resulting smoothed<br />
Reuleaux triangle is equal to the sum of the two radii so employed [125].<br />
Property 57 (Least-Area Rotor). The least-area rotor for an equilateral<br />
triangle is formed from two 60 ◦ circular arcs with radius equal to the altitude<br />
of the triangle [291] (which coincides with the length of the rotor [322]) (Figure<br />
2.38). (The incircle is the greatest-area rotor [39].)<br />
As it rotates, its corners trace the entire boundary of the triangle without<br />
rounding of corners [125], although it must slip as it rolls [291].
Mathematical Properties 57<br />
Figure 2.38: Least-Area Rotor Figure 2.39: Triangular Rotor<br />
Property 58 (Equilateral Triangular Rotor). Obviously, the equilateral<br />
triangle is a rotor for a circle. Yet, it can also be a rotor for a noncircular<br />
cylinder.<br />
Figure 2.39 shows the curve given parametrically as<br />
x(t) = cost + .1 cos 3t; y(t) = sint + .1 sin 3t,<br />
within which rotates an equilateral triangular piston [291]. (See Application<br />
26.)<br />
Figure 2.40: Kakeya Needle Problem [125]
58 Mathematical Properties<br />
Property 59 (Kakeya Needle Problem). The convex plane figure of least<br />
area in which a line segment of length 1 can be rotated through 180 ◦ , returning<br />
to its original position with reversed orientation, is an equilateral triangle with<br />
altitude 1 (area 1/ √ 3) [125].<br />
The required rotation is illustrated in Figure 2.40. If the constraint of<br />
convexity is removed then there is no such plane figure of smallest area [14,<br />
pp. 99-101]!<br />
(a) (b) (c)<br />
Figure 2.41: Equilateral Triangular Fractals: (a) Koch Snowflake [133]. (b)<br />
Sierpinski Gasket [314]. (c) Golden Triangle Fractal [314].<br />
Property 60 (Equilateral Triangular Fractals). The equilateral triangle<br />
is eminently suited for the construction of fractals.<br />
In Figure 2.41(a), the Koch Snowflake is constructed by successively replacing<br />
the middle third of each edge by the other two sides of an equilateral<br />
triangle [100]. Although the perimeter is infinite, the area bounded by the<br />
curve is exactly 8 that of the initial triangle [133] and its fractal dimension<br />
5<br />
is log 4/ log 3 ≈ 1.2619 [322]. (The “anti-snowflake” curve is obtained if the<br />
appended equilateral triangles are turned inwards instead of outwards, with<br />
area 2<br />
5<br />
of the original triangle [66].) In Figure 2.41(b), the Sierpinski Gasket is<br />
obtained by repeatedly removing (inverted) equilateral triangles from an initial<br />
equilateral triangle [100]. Its fractal dimension is equal to log 3/ log 2 ≈ 1.5850.<br />
In Figure 2.41(c), the Golden Triangle Fractal is generated from an initial<br />
equilateral triangle by successively adding to any free corner at each stage an<br />
equilateral triangle scaled in size by 1<br />
φ where φ = 1+√5 is the golden section<br />
2<br />
[314]. Its fractal dimension is equal to log 2/ log φ ≈ 1.4404.<br />
Property 61 (Pascal’s Triangle). In 1654, Pascal published Traité du triangle<br />
arithmétique wherein he intensively studied the Arithmetical Triangle<br />
(PAT) shown in Figure 2.42(a), where each entry is the sum of its northwestern<br />
and northeastern neighbors [93].
Mathematical Properties 59<br />
(a) (b) (c)<br />
Figure 2.42: Pascal’s Triangle (PT): (a) Arithmetical PT (1654). (b) Chinese<br />
PT (1303) . (c) Fractal PT (1965).<br />
It is in fact much older, appearing in the works of Mathematicians throughout<br />
Persia, India and China. One such instance is shown in Figure 2.42(b),<br />
which is taken from Chu shih-chieh’s Precious Mirror of the Four Elements of<br />
1303. Can you find the mistake which is buried therein? (Hint: Look in Row<br />
7, where rows are numbered beginning at 0.)<br />
The mathematical treasures hidden within PAT are truly staggering, e.g.<br />
the binomial coefficients and the triangular numbers, but we will focus our<br />
attention on the following gem. Consider what happens when odd numbers<br />
in PAT are darkened and even numbers are left blank. Extending PAT to<br />
infinitely many rows and reducing the scale by one-half each time the number<br />
of rows is doubled produces the previously encountered fractal, Sierpinski’s<br />
Gasket (Figure 2.42(c)) [127]!<br />
Property 62 (The Chaos Game). Equilateral triangular patterns can emerge<br />
from chaotic processes.<br />
Choose any point lying within an equilateral triangle, the vertices of which<br />
are labeled 1 thru 3, and mark it with a small dot. Roll a cubic die to produce<br />
a number n and set i = (n mod 3) + 1. Generate a new point located at<br />
the midpoint of the segment connecting the previous dot with vertex i and<br />
mark it with a small dot. Iterate this process k times always connecting the<br />
most recent dot with the latest randomly generated vertex. The results for<br />
(a) k = 100, (b) k = 500, (c) k = 1, 000 and (d) k = 10, 000 are plotted in<br />
Figure 2.43. Voila, Sierpinski’s Gasket emerges from this chaotic process [239,<br />
Chapter 6]!
60 Mathematical Properties<br />
Figure 2.43: Chaos Game [239]<br />
Figure 2.44: Equilateral Lattice [172]
Mathematical Properties 61<br />
Property 63 (Equilateral Lattice). Let c denote the minimum distance<br />
between two points in a unit lattice, i.e. one�constructed from an arbitrary<br />
2<br />
parallelogram of unit area [172]. Then, c ≤ √3 , and this upper bound is<br />
achieved by the lattice generated by a parallelogram that is composed of two<br />
equilateral triangles (i.e. by a “regular rhombus”).<br />
Moreover, for a given value of c, this lattice has the smallest possible generating<br />
parallelogram. Asymptotically, of all lattices with a given c, the lattice<br />
composed of equilateral triangles has the greatest number of points in a given<br />
large region. Finally, the lattice of equilateral triangles gives rise to the densest<br />
≈ .907.<br />
packing of circles of radius c<br />
2<br />
(Figure 2.44) with density D = π<br />
2 √ 3<br />
Property 64 (No Equilateral Triangles on a Chess Board). There is<br />
no equilateral triangle whose vertices are plane lattice points [96].<br />
This was one of 24 theorems proposed in a survey on The Most Beautiful<br />
Theorems in Mathematics [321]. It came in at Number 19. In general,<br />
a triangle is embeddable in Z 2 if and only if all of its angles have rational<br />
tangents [17]. Of course, the equilateral triangle is embeddable in Z n (n ≥ 3):<br />
{(1, 0, 0, . . . ), (0, 1, 0, . . . ), (0, 0, 1, . . . )}. M. J. Beeson has provided a complete<br />
characterization of the triangles embeddable in Z n for each n [17].<br />
Figure 2.45: Equilateral Triangular Mosaic [199]<br />
Property 65 (Regular Tessellations of the Plane). The only regular tessellations<br />
of the plane by polygons of the same kind meeting only at a vertex<br />
are provided by equilateral triangles, squares and regular hexagons. If a vertex<br />
of one polygon is allowed to lie on the side of another then the only such tessellations<br />
are afforded by equilateral triangles (Figure 2.45) and squares [199,<br />
pp. 199-202].
62 Mathematical Properties<br />
Figure 2.46: Dirichlet Duality [232]<br />
Property 66 (Triangular and Hexagonal Lattice Duality). The Dirichlet<br />
(Voronoi) region associated with a lattice point is the set of all points closer<br />
to it than to any other lattice point [232]. The regular hexagonal and equilateral<br />
triangular lattices are dual to one another in the sense that they are each<br />
other’s Dirichlet tessellation (Voronoi diagram) (Figure 2.46) [209].<br />
Property 67 (From Tessellations to Fractals). An infinite sequence of<br />
tessellating shapes based upon the equilateral triangle may give rise to a limiting<br />
fractal pattern [234].<br />
Begin by dissecting an equilateral triangle, the Level 0 tile, into sixteen<br />
smaller copies, three of which are shown in Figure 2.47(a), by subdividing<br />
each edge into fourths. Then, hinging these three as shown in Figure 2.47(b)<br />
and rotating them counterclockwise as in Figure 2.47(c) produces the Level 1<br />
tile of Figure 2.47(d). The same tripartite process of dissect, hinge and rotate<br />
may be applied to this Level 1 tile to produce the Level 2 tile of Figure 2.48.<br />
The dissection process is illustrated in Figure 2.49 and the net result of hinging<br />
and rotating is on display in Figure 2.50. This process of dissection into 16<br />
congruent pieces followed by hinge-rotation followed by a size reduction of onefourth<br />
(in length) leads to an infinite cascade of shapes (Level 3 is shown in<br />
Figure 2.51) which lead, in the limit, to a self-similar fractal shape. Other Level<br />
0 shapes based upon the equilateral triangle and square may be investigated<br />
[234].
Mathematical Properties 63<br />
(a)<br />
(b) (c) (d)<br />
Figure 2.47: Level 1: (a) Dissect. (b) Hinge. (c) Rotate. (d) Tile. [234]<br />
Figure 2.48: Level 2 Figure 2.49: Level 1 Tessellation<br />
Figure 2.50: Level 2 Reprise Figure 2.51: Level 3
64 Mathematical Properties<br />
(a)<br />
(b)<br />
Figure 2.52: Propeller Theorem: (a) Symmetric Propellers. (b) Asymmetric<br />
Propellers. (c) Triangular Hub. [138]<br />
Property 68 (The Propeller Theorem). The Propeller Theorem states<br />
that the midpoints of the three chords connecting three congruent equilateral<br />
triangles which are joined at a vertex lie at the vertices of an equilateral triangle<br />
(Figure 2.52(a)) [138].<br />
In fact, the triangular propellers may even touch along an edge or overlap.<br />
The Asymmetric Propeller Theorem states that the three equilateral triangles<br />
need not be congruent (Figure 2.52(b)). The Generalized Asymmetric Propeller<br />
Theorem states the propellers need not meet at a point but may meet<br />
at the vertices of an equilateral triangle (Figure 2.52(c)). Finally, the General<br />
Generalized Asymmetric Propeller Theorem states the the propellers need not<br />
even be equilateral, as long as they are all similar triangles! If they do not<br />
meet at a point then they must meet at the vertices of a fourth similar triangle<br />
and the vertices of the triangular hub must meet the propellers at their<br />
corresponding corners [138].<br />
Figure 2.53: Tetrahedral Geodesics [305]<br />
(c)
Mathematical Properties 65<br />
Property 69 (Tetrahedral Geodesics). A geodesic is a generalization of<br />
a straight line which, in the presence of a metric, is defined to be the (locally)<br />
shortest path between two points as measured along the surface [305].<br />
For example, the tetrahedron MNPQ of Figure 2.53(a) can be transformed<br />
into the equivalent planar net of Figure 2.54(a) by cutting the surface of the<br />
tetrahedron along the edges MN, MP and MQ, rotating the triangle MNP<br />
about the edge NP until it is in the same plane as the triangle NPQ, and<br />
then performing the analogous operations on the triangles MPQ and MQN.<br />
Now, in Figure 2.53(a), let A be the point of triangle MNP which lies one<br />
third of the way up from N on the perpendicular from N to MP; and let B be<br />
the corresponding point in triangle MPQ. Then, we obtain the geodesic (and<br />
at the same time the shortest) line connecting A and B on the surface of the<br />
tetrahedron, shown in Figure 2.53(b), by simply drawing the dashed straight<br />
line connecting A to B in Figure 2.54(a). (The length of this geodesic is equal<br />
to the length of the edge NQ of the tetrahedron, which we take to be 1.)<br />
Figure 2.53(c) shows another geodesic connecting these same two points but<br />
which is longer. From Figure 2.54(b), the length of this geodesic is equal to<br />
2 √ 3/3 ≈ 1.1547. This net is obtained by cutting the surface of the tetrahedron<br />
along the edges MQ, MN and NP.<br />
Figure 2.54: Transformation to Planar Net [305]<br />
Property 70 (Malfatti’s Problem). In 1803, G. Malfatti proposed the problem<br />
of constructing three circles within a given triangle each of which is tangent<br />
to the other two and also to two sides of the triangle, as on the right of Figure<br />
2.55 [231]. He assumed that this would provide a solution to the “marble<br />
problem”: to cut out from a triangular prism, made of marble, three circular<br />
columns of the greatest possible volume (i.e., wasting the least possible amount<br />
of marble).
66 Mathematical Properties<br />
Figure 2.55: Malfatti’s Problem [231]<br />
In 1826, J. Steiner published, without proof, a purely geometrical construction<br />
while, in 1853, K. H. Schellbach published an elementary analytical<br />
solution [84]. Then, in 1929, H. Lob and H. W. Richmond showed that, for<br />
the equilateral triangle, the Malfatti circles did not solve the marble problem.<br />
≈ 0.739 of the<br />
The correct solution, shown on the left of Figure 2.55, fills 11π<br />
27 √ 3<br />
triangle area while the Malfatti circles occupy only π √ 3<br />
(1+ √ 3) 2 ≈ 0.729 of that area<br />
[2]. Whereas there is only this tiny 1% discrepancy for the equilateral triangle,<br />
in 1965, H. Eves pointed out that if the triangle is long and thin then the<br />
discrepancy can approach 2:1 [231]. In 1967, M. Goldberg demonstrated that<br />
the Malfatti circles never provide the solution to the marble problem [322].<br />
Finally, in 1992, V. A. Zalgaller and G. A. Los gave a complete solution to the<br />
marble problem.<br />
(a)<br />
(b)<br />
Figure 2.56: Group of Symmetries [108]<br />
Property 71 (Group of Symmetries). The equilateral triangle and the regular<br />
tiling of the plane which it generates, {3}, possess three lines of reflectional<br />
symmetry and three degrees of rotational symmetry as can be seen in Figure<br />
2.56 [327].<br />
(c)
Mathematical Properties 67<br />
These isometries form the dihedral group of order 6, D3, and its group<br />
table also appears in Figure 2.56 where ρi stands for rotations and µi for<br />
mirror images in angle bisectors [108].<br />
(a) (b)<br />
Figure 2.57: Color Symmetry: (a) Fundamental Domain. (b) Perfect Coloring.<br />
[190]<br />
Property 72 (Perfect Coloring). Starting from a fundamental domain (denoted<br />
by I in Figure 2.57(a)), a symmetry tiling of the equilateral triangular tile<br />
may be generated by operating on it with members of the group of isometries,<br />
D3.<br />
The five replicas so generated are labeled by the element of D3 that maps<br />
it from the fundamental domain where, in our previous notation, R1 = µ1,<br />
R2 = µ3, R3 = µ2, S = ρ2, S 2 = ρ1. A symmetry of a tiling that employs only<br />
two colors, say black and white, is called a two-color symmetry whenever each<br />
symmetry of the uncolored tiling either transforms all black tiles to black tiles<br />
and all white tiles to white tiles or transforms all black tiles to white tiles and all<br />
white tiles to black tiles. When every symmetry of the uncolored tiling is also<br />
a two-color symmetry, the coloring is called perfect. I.e., in a perfect coloring,<br />
each symmetry of the uncolored tiling simply induces a permutation of the<br />
colors in the colored tiling. A perfect coloring of the equilateral triangular tile<br />
is on display in Figure 2.57(b) [190].<br />
Property 73 (Fibonacci Triangle). Shade in a regular equilateral triangular<br />
lattice as shown in Figure 2.58 so that a rhombus (light) lies under each<br />
trapezoid (dark) and vice versa. Then, the sides of successive rhombi form a<br />
Fibonacci sequence (1,1,2,3,5,8,...) and the top, sides and base of each trapezoid<br />
are three consecutive Fibonacci numbers [315].
68 Mathematical Properties<br />
Figure 2.58: Fibonacci Triangle [315]<br />
Figure 2.59: Golden Ratio [253]
Mathematical Properties 69<br />
Property 74 (Golden Ratio). With reference to Figure 2.59 left, let ABC<br />
be an equilateral triangle inscribed in a circle, let L and M be the midpoints of<br />
AB and AC, respectively, and let LM meet the circle at X and Y as shown;<br />
then, LM/MY = φ where φ = (1+ √ 5)/2 is the golden ratio (limiting ratio of<br />
successive Fibonacci numbers) [253]. The pleasing design in Figure 2.59 right<br />
may be readily produced, in which the ratio of the sides of the larger to the<br />
smaller triangles is equal to φ.<br />
This intriguing result was first observed by George Odom, a resident of the<br />
Hudson River Psychiatric Center, in the early 1980s [254, p. 10]. Upon communicating<br />
it to the late H. S. M. Coxeter, it was submitted to the American<br />
Mathematical Monthly as Problem E3007, Vol. 90 (1983), p. 482 with the<br />
solution appearing in Vol. 93 (1986), p. 572.<br />
(a) (b) (c)<br />
Figure 2.60: (a) Bination of Equilateral Triangle. (b) Male Equilateral Spiral.<br />
(c) Female Equilateral Spiral. [83]<br />
Property 75 (Equilateral Spirals). E. P. Doolan has introduced the notion<br />
of equilateral spirals [83].<br />
Successive subdivision of an equilateral triangle by systematically connecting<br />
edge midpoints, as portrayed in Figure 2.60(a), is called (clockwise) bination.<br />
(Gazalé [139, p. 111] calls the resulting configuration a “whorled<br />
equilateral triangle”.) Retention of the edge counterclockwise to the new edge<br />
produced at each stage produces a (clockwise) male equilateral spiral (Figure<br />
2.60(b)). Replacement of each edge of such a male equilateral spiral by the<br />
arc of the circumcircle subtended by that edge produces the corresponding<br />
(clockwise) female equilateral spiral (Figure 2.60(c)).<br />
Doolan has shown that the female equilateral spiral is in fact C 1 . Moreover,<br />
both the male and female equilateral spirals are geometric in the sense that,<br />
for a fixed radius emanating from the spiral center, the intersections with the<br />
spiral are at a constant angle. Note that this is distinct from equiangularity in<br />
that this angle is different for different radii. In addition, he has investigated<br />
the “sacred geometry” of these equilateral spirals and shown how to construct<br />
them with only ruler and compass.
70 Mathematical Properties<br />
(a)<br />
(b)<br />
Figure 2.61: Padovan Spirals: (a) Padovan Whorl [138]. (b) Inner Spiral [293].<br />
(c) Outer Spiral [139].<br />
Property 76 (Padovan Spirals). Reminiscent of the Fibonacci sequence,<br />
the Padovan sequence is defined as 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, . . ., where<br />
each number is the sum of the second and third numbers preceding it [138].<br />
The �ratio<br />
of succesive � terms of this sequence approaches the plastic number,<br />
p = 3<br />
1<br />
2<br />
+ 1<br />
6<br />
� 23<br />
3<br />
+ 3<br />
1<br />
2<br />
− 1<br />
6<br />
� 23<br />
3<br />
(c)<br />
≈ 1.324718, which is the real solution of<br />
p 3 = p + 1 [139]. The Padovan triangular whorl [139] is formed from the<br />
Padovan sequence as shown in Figure 2.61(a). Note that each triangle shares<br />
a side with two others thereby giving a visual proof that the Padovan sequence<br />
also satisfies the recurrence relation pn = pn−1 + pn−5. If one-third of a circle<br />
is inscribed in each triangle, the arcs form the elegant spiral of Figure 2.61(b)<br />
which is a good approximation to a logarithmic spiral [293]. Beginning with a<br />
gnomon composed of a “plastic pentagon” (ABCDE in Figure 2.61(c)) with<br />
sides in the ratio 1 : p : p 2 : p 3 : p 4 , if we add equilateral triangles that grow in<br />
size by a factor of p, then a truly logarithmic spiral is so obtained [139].<br />
Property 77 (Perfect Triangulation). In 1948, W. T. Tutte proved that it<br />
is impossible to dissect an equilateral triangle into smaller equilateral triangles<br />
all of different sizes (orientation ignored) [309]. However, if we distinguish between<br />
upwardly and downwardly oriented triangles then such a “perfect” tiling<br />
is indeed possible [310].<br />
Figure 2.62, where the numbers indicate the size (side length) of the components<br />
in units of a primitive equilateral triangle, shows a dissection into 15<br />
pieces, which is believed to be the lowest order possible. E. Buchman [35] has<br />
extended Tutte’s method of proof to conclude that no planar convex region<br />
can be tiled by unequal equilateral triangles. Moreover, he has shown that
Mathematical Properties 71<br />
Figure 2.62: Perfect Triangulation [310]<br />
any nonequilateral triangle can be tiled by smaller unequal triangles similar to<br />
itself.<br />
Figure 2.63: Convex Tilings [294]<br />
Property 78 (Convex Tilings). In 1996, R. T. Wainwright posed the question:<br />
What is the largest convex area that can be tiled with a given number of<br />
equilateral triangles whose sides are integers, where, to avoid trivially scaling<br />
up the size of a given tiling, the sizes of the tiles are further constrained to<br />
have no overall common divisor [294]?
72 Mathematical Properties<br />
The best known such tiling with 15 equilateral triangles is shown in Figure<br />
2.63 and has an area of 4,782 etus which is a considerable improvement over<br />
the minimum order perfect triangulation with area of 1,374 etus.<br />
Figure 2.64: Partridge Tiling [162]<br />
Property 79 (Partridge Tiling). A partridge tiling of order n of an equilateral<br />
triangle is composed of 1 equilateral triangle of side 1, 2 equilateral<br />
triangles of side 2, and so on, up to n equilateral triangles of side n [162].<br />
The partridge number of the equilateral triangle is defined to be the smallest<br />
value of n for which such a tiling is possible. W. Marshall discovered the<br />
partridge tiling of Figure 2.64 and P. Hamlyn showed that this is indeed the<br />
smallest possible, and so the equilateral triangle has a partridge number of 9<br />
[162].<br />
Figure 2.65: Partition of an Equilateral Triangle [149]
Mathematical Properties 73<br />
Property 80 (Partitions of an Equilateral Triangle). Let T denote a<br />
closed unit equilateral triangle. For a fixed integer n, let dn denote the infimum<br />
of all those x for which it is possible to partition T into n subsets, each subset<br />
having a diameter not exceeding x. Recall that the diameter of a plane set<br />
A is given by d(A) = sup a,b∈A ρ(a, b) where ρ(a, b) is the Euclidean distance<br />
between a and b. R. L. Graham [149] has determined dn for 1 ≤ n ≤ 15.<br />
Figure 2.65 gives an elegant partition of T into 15 sets each having diameter<br />
d15 = 1/(1 + 2 √ 3).<br />
Property 81 (Dissecting a Polygon into Nearly-Equilateral Triangles).<br />
Every polygon can be dissected into acute triangles. On the other hand,<br />
a polygon P can be dissected into equilateral triangles with (interior) angles<br />
arbitrarily close to π/3 radians if and only if all of the angles of P are multiples<br />
of π/3. For every other polygon, there is a limit to how close it can come<br />
to being dissected into equilateral triangles [64, pp. 89-90].<br />
Figure 2.66: Regular Simplex [60]<br />
Property 82 (Regular Simplex). A regular simplex is a generalization of<br />
the equilateral triangle to Euclidean spaces of arbitrary dimension [60]. Given<br />
a set of n + 1 points in R n which are pairwise equidistant (distance = d), an<br />
n-simplex is their convex hull.<br />
A 2-simplex is an equilateral triangle, a 3-simplex is a regular tetrahedron<br />
(shown in Figure 2.66) , a 4-simplex is a regular pentatope and, in general, an<br />
n-simplex is a regular polytope [60]. The convex hull of any nonempty proper<br />
subset of the given n+1 mutually equidistant points is itself a regular simplex<br />
of lower dimension called an m-face. The n + 1 0-faces are called vertices, the<br />
n(n+1)<br />
1-faces are called edges, and the n + 1 (n − 1)-faces are called facets.<br />
2<br />
� �<br />
n + 1<br />
In general, the number of m-faces is equal to and so may be found<br />
m + 1
74 Mathematical Properties<br />
in column m + 1 of row n + 1 of Pascal’s triangle. A regular n-simplex may<br />
be constructed from a regular (n − 1)-simplex by connecting a new vertex to<br />
all of the original vertices by an edge of length d. A regular n-simplex is so<br />
named because it is the simplest regular polytope in n dimensions.<br />
(a) (b)<br />
Figure 2.67: Non-Euclidean Equilateral Triangles: (a) Spherical. (b) Hyperbolic.<br />
[318]<br />
Property 83 (Non-Euclidean Equilateral Triangles). Figure 2.67 displays<br />
examples of non-Euclidean equilateral triangles.<br />
On a sphere, the sum of the angles in any triangle always exceeds π radians.<br />
On the unit sphere (with constant curvature +1 and area 4π), the area of<br />
an equilateral triangle, A, and one of its three interior angles, θ, satisfy the<br />
relation A = 3θ − π [318]. Thus, limA→0 θ = π/3. The largest equilateral<br />
triangle, corresponding to θ = π, encloses a hemisphere with its three vertices<br />
equally spaced along a great circle. I.e., limA→2π θ = π. Figure 2.67(a) shows a<br />
spherical tessellation by the 20 equilateral triangles, corresponding to θ = 2π/5<br />
and A = π/5, associated with an inscribed icosahedron. On a hyperbolic plane,<br />
the sum of the angles in any triangle is always smaller than π radians. On<br />
the standard hyperbolic plane, H 2 , (with constant curvature -1), the area and<br />
interior angle of an equilateral triangle satisfy the relation A = π − 3θ [318].<br />
Once again, limA→0 θ = π/3. Observe the peculiar fact that an equilateral<br />
triangle in the standard hyperbolic plane (which is unbounded!) can never<br />
have an area exceeding π. This seeming conundrum is resolved by reference to<br />
Figure 2.67(b) where a sequence of successively larger equilateral hyperbolic<br />
triangles are represented in the Euclidean plane. Note that, as the sides of the<br />
triangle become unbounded, the angles approach zero while the area remains<br />
bounded. I.e., limA→π θ = 0.
Mathematical Properties 75<br />
(a)<br />
(b) (c)<br />
Figure 2.68: The Hyperbolic Plane: (a) Embedded Patch [302]. (b) Poincaré<br />
Disk. (c) Thurston Model [20]. [138]<br />
Property 84 (The Hyperbolic Plane). The crochet model of Figure 2.68(a)<br />
displays a patch of H 2 embedded in R 3 [302].<br />
As shown in Figure 2.68(b), it may also be modeled by the Poincaré disk<br />
whose geodesics are either diameters or circular arcs orthogonal to the boundary<br />
[33]. In this figure, the disk has been tiled by equilateral hyperbolic triangles<br />
meeting 7 at a vertex. This tiling ultimately led to the Thurston model of<br />
the hyperbolic plane shown in Figure 2.68(c) [20]. In this model, 7 Euclidean<br />
equilateral triangles are taped together at each vertex so as to provide novices<br />
with an intuitive feeling for hyperbolic space [318]. However, it is important<br />
to note that the Thurston model can be misleading if it is not kept in mind<br />
that it is but a qualitative approximation to H 2 [20].<br />
Property 85 (The Minkowski Plane). In 1975, L. M. Kelly proved the<br />
conjecture of M. M. Day to the effect that a Minkowski plane with a regular<br />
dodecagon as unit circle satisfies the norm identity [192]:<br />
||x|| = ||y|| = ||x − y|| = 1 ⇒ ||x + y|| = √ 3.<br />
Stated more geometrically, the medians of an equilateral triangle of side<br />
length s are of length √ 3 · s just as they are in the Euclidean plane. Midpoint<br />
2<br />
in this context is interpreted vectorially rather than metrically.<br />
Property 86 (Mappings Preserving Equilateral Triangles). Sikorska<br />
and Szostok [281] have shown that if E is a finite-dimensional Euclidean space<br />
with dim E ≥ 2 then f : E → E is measurable and preserves equilateral<br />
triangles implies that it is a similarity transformation (an isometry multiplied<br />
by a positive constant).<br />
Since such a similarity transformation preserves every shape, this may be<br />
paraphrased to say that if a measurable function preserves a single shape, i.e.<br />
that of the equilateral triangle, then it preserves all shapes. In [282], they<br />
extend this result to normed linear spaces.
76 Mathematical Properties<br />
Figure 2.69: Delahaye Product<br />
Property 87 (Delahaye Product). In his Arithmetica Infinitorum (1655),<br />
John Wallis presented the infinite product representation<br />
π<br />
2<br />
2 2 4 4 6 6 8 8<br />
= · · · · · · · · · · .<br />
1 3 3 5 5 7 7 9<br />
In 1997, Jean-Paul Delahaye [74, p. 205] presented the related infinite product<br />
2π<br />
3 √ 3<br />
3 3 6 6 9 9 12 12<br />
= · · · · · · · · · · .<br />
2 4 5 7 8 10 11 13<br />
The presence of π together with √ 3 suggests that a relationship between the<br />
circle and the equilateral triangle may be hidden within this formula.<br />
We may disentangle these threads as follows. The left-hand-side expression,<br />
pR 2π = p 3 √ , is the ratio of the perimeter of the circumcircle to the perime-<br />
3<br />
ter of the equilateral triangle. Introducing the scaling parameters σk √ :=<br />
(3k−1)(3k+1)<br />
, Delahaye’s product may be rewritten as<br />
3k<br />
pR · σ<br />
lim<br />
k→∞<br />
2 1 · σ2 2 · · ·σ 2 k · · ·<br />
p<br />
= 1.<br />
Thus, if we successively shrink the circumcircle by multiplying its radius by<br />
the factors, σ2 k (k = 1, . . . , ∞), then the resulting circles approach a limiting<br />
position where the perimeter of the circle coincides with that of the equilateral<br />
triangle (Figure 2.69).
Mathematical Properties 77<br />
Property 88 (Grunsky-Motzkin-Schoenberg Formula). Suppose that<br />
f(z) is analytic on the equilateral triangle, T, with vertices at 1, w, w 2 where<br />
w := exp (2πı/3). Then [69, p. 129],<br />
� �<br />
f<br />
T<br />
′′ (z) dxdy =<br />
√ 3<br />
2 · [f(1) + wf(w) + w2 f(w 2 )].<br />
While this chapter has certainly made a strong case for the mathematical<br />
richness associated with the equilateral triangle, it runs the risk of leaving the<br />
reader with the impression that it has only theoretical and aesthetic value or,<br />
at best, is useful only within Mathematics itself. Nothing could be further<br />
from the truth! In the next chapter, I will present a sampling of applications<br />
of the equilateral triangle which have been selected to provide a feel for the<br />
diversity of practical uses of the equilateral triangle for comprehending the<br />
world about us that the human race has uncovered (so far).
Chapter 3<br />
Applications of the Equilateral<br />
Triangle<br />
Figure 3.1: Equilateral Triangle Method in Surveying [34]<br />
Application 1 (Surveying). In surveying, the Equilateral Triangle Method<br />
[34] is used to measure around obstacles.<br />
With reference to Figure 3.1, point B is set on the transit line as near the<br />
obstacle as practicable but so that a line BC at 60 ◦ with the transit line can be<br />
run out. The instrument is then set up at B, backsighted on A, and an angle of<br />
120 ◦ laid off. The line BC is made long enough so that, when the instrument<br />
is set up at C and 60 ◦ is laid off from it, CD will lie outside the obstacle. BC<br />
is measured and CD is made equal to BC. If now the instrument is set up at<br />
D and angle ∠CDE laid off equal to 120 ◦ then the line DE is the continuation<br />
of the original transit line and BD = BC.<br />
78
Applications 79<br />
Figure 3.2: North American Satellite Triangulation Network [53]<br />
Figure 3.3: GPS Antenna [274]
80 Applications<br />
Application 2 (Satellite Geodesy). In satellite geodesy (the forerunner to<br />
today’s GPS)[53], an equilateral triangle on the Earth (comprised of stations<br />
at Aberdeen, MD; Chandler, MN; and Greenville, MS) with sides roughly 900<br />
miles long was first used in 1962 to verify the accuracy of the satellite triangulation<br />
concept.<br />
With reference to Figure 3.2, the U.S. Coast and Geodetic Survey used<br />
three specially designed ballistic cameras with associated electronic time synchronization<br />
systems to track the motion of the NASA ECHO I communications<br />
satellite. These three observation stations were tied to the existing<br />
triangulation network for the test. After several months of observations, it<br />
was concluded that this process offered precision comparable to, or better<br />
than, the existing conventional triangulation network. This allowed them to<br />
use this process to strengthen the North American Network which includes<br />
the continental United States and Alaska via Canada as well as the islands of<br />
Antigua and Bermuda.<br />
Application 3 (GPS Antenna). An equilateral triangular receiving antenna<br />
can be used in the Global Positioning System (GPS) [274].<br />
With reference to Figure 3.3, C. Scott has constructed a GPS receiving<br />
antenna based upon an equilateral triangular blade monopole design which<br />
approaches the broadband characteristics of a conical monopole. This in turn<br />
gives very broad resonance and reasonable impedance matching. As the aircraft<br />
receiver is not portable, this antenna is suitable for interfacing and debugging<br />
in the laboratory.<br />
Figure 3.4: “Bat’s Ear” Antenna [106]
Applications 81<br />
Application 4 (Biomimetic “Bat’s Ear” Antenna). A biomimetic antenna<br />
in the shape of a bat’s ear may be constructed from an equilateral triangular<br />
conducting plate that is curved and the base electrically connected to a<br />
circular ground plane with a central monopole element [106].<br />
With reference to Figure 3.4, J. A. Flint [106] has shown that, for certain<br />
frequencies, this yields a higher gain and a radiation pattern with lower side<br />
lobes than the equivalent circular ground mounted monopole and that a good<br />
match can be retained at the coaxial input.<br />
(a) (b)<br />
Figure 3.5: (a) Principle of the Equilateral Triangle in Electrocardiography.<br />
(b) Graphical Determination of the Electrical Axis of the Heart. [45]<br />
Application 5 (Principle of the Equilateral Triangle in Electrocardiography).<br />
Einthoven’s Triangle of Electrocardiography (Figure 3.5(a)), with<br />
vertices comprised of electrodes located on the left arm (LA), right arm (RA)<br />
and left leg (LL), is used to determine the electrical axis of the heart [45].<br />
Normally, this electrical axis is oriented in a right shoulder to left leg direction.<br />
Any significant deviation of the electrical axis from this orientation can<br />
indicate ventricular hypertrophy (straining).<br />
The electrical activity of the heart can be described by the movement of an<br />
electrical dipole consisting of a negative and a positive charge separated by a<br />
variable distance. The directed line segment joining these two charges is called<br />
the cardiac vector. Its magnitude and direction can be described by three<br />
vectors along the edges of an equilateral triangle, each vector representing the<br />
potential difference, ei, across electrical leads connecting the electrodes (e1:<br />
lead 1 from RA to LA; e2: lead 2 from RA to LL; e3: lead 3 from LA to LL).
82 Applications<br />
The ei are the projections of the cardiac vector onto the three sides of the<br />
Einthoven Triangle and e1 − e2 + e3 = 0. Furthermore,<br />
tanα = 2e2 − e1<br />
√ =<br />
3 2e3 + e1<br />
e1<br />
√ =<br />
e1 3 e2 + e3<br />
(e2 − e3) √ 3 ,<br />
where α is the angle of inclination of the electrical axis of the heart. Along the<br />
corresponding edge of the triangle, a point a distance ei (measured from the<br />
EKG) from the midpoint is marked off. The perpendiculars emanating from<br />
these three points meet at a point inside the triangle. The vector from the<br />
center of the Einthoven Triangle to this point of intersection represents the<br />
cardiac vector. Its angle of inclination is then easily read from the graphical<br />
device shown in Figure 3.5(b).<br />
Figure 3.6: Human Elbow [88]<br />
Application 6 (Human Elbow). Three bony landmarks of the human elbow<br />
- the medial epicondyle, the lateral epicondoyle, and the apex of the olecranon<br />
- form an approximate equilateral triangle when the elbow is flexed 90 ◦ , and a<br />
straight line when the elbow is in extension (Figure 3.6) [88].<br />
Figure 3.7: Lagrange’s Equilateral Triangle Solution [171]
Applications 83<br />
Application 7 (Lagrange’s Equilateral Triangle Solution (Three-Body<br />
Problem)). The three-body problem requires the solution of the equations of<br />
motion of three mutually attracting masses confined to a plane. One of the few<br />
known analytical solutions is Lagrange’s Equilateral Triangle Solution [171].<br />
As illustrated in Figure 3.7 (m1 : m2 : m3 = 1 : 2 : 3), the particles<br />
sit at the vertices of an equilateral triangle as this triangle changes size and<br />
rotates. Each particle follows an elliptical path of the same eccentricity but<br />
oriented at different angles with their common center of mass located at a focal<br />
point of all three orbits. The motion is periodic with the same period for all<br />
three particles. This solution is stable if and only if one of the three masses is<br />
much greater than the other two. However, very special initial conditions are<br />
required for such a configuration.<br />
Figure 3.8: Lagrangian Points [1]<br />
Application 8 (Lagrangian Points (Restricted Three-Body Problem)).<br />
In the circular restricted three-body problem, one of the three masses is taken<br />
to be negligible while the other two masses assume circular orbits about their<br />
center of mass.<br />
There are five points (Lagrangian points, L-points, libration points) where<br />
the gravitational forces of the two large bodies exactly balance the centrifugal<br />
force felt by the small body [1]. An object placed at one of these points<br />
would remain in the same position relative to the other two. Points L4 and<br />
L5 are located at the vertices of equilateral triangles with base connecting<br />
the two large masses; L4 lies 60 ◦ ahead and L5 lies 60 ◦ behind as illustrated<br />
in Figure 3.8. These two Lagrangian Points are (conditionally) stable under<br />
small perturbations so that objects tend to accumulate in the vicinity of these<br />
points. The so-called Trojan asteroids are located at the L4 and L5 points<br />
of the Sun-Jupiter system. Furthermore, the Saturnian moon Tethys has two<br />
smaller moons, Telesto and Calypso, at its L4 and L5 points while the Saturn-<br />
Dione L4 and L5 points hold the small moons Helene and Polydeuces [296, p.<br />
222].
84 Applications<br />
(a)<br />
Figure 3.9: (a) LISA (Laser Interferometer Space Antenna). (b) The LISA<br />
Constellation’s Heliocentric Orbit. [117]<br />
Application 9 (LISA and Gravitational Waves). Launching in 2020 at<br />
the earliest, LISA (Laser Interferometer Space Antenna) [117] will overtake<br />
the Large Hadron Collider as the world’s largest scientific instrument.<br />
Einstein’s General Theory of Relativity predicts the presence of gravitational<br />
waves produced by massive objects, such as black holes and neutron<br />
stars, but they are believed to be so weak that they have yet to be detected.<br />
This joint project of ESA and NASA to search for gravitational waves will<br />
consist of three spacecraft arranged in an equilateral triangle, 5 million kilometers<br />
(3.1 million miles or 1/30 of the distance to the Sun) on each side, that<br />
will tumble around the Sun 20 ◦ behind Earth in its orbit (Figure 3.9(a)). The<br />
natural free-fall orbits of the three spacecraft around the Sun will maintain this<br />
triangular formation. The plane of the LISA triangle will be inclined at 60 ◦<br />
to the ecliptic, and the triangle will appear to rotate once around its center in<br />
the course of a year’s revolution around the Sun (Figure 3.9(b)). Each spacecraft<br />
will house a pair of free-floating cubes made of a gold-platinum alloy and<br />
the distance between the cubes in different spacecraft will be monitored using<br />
highly accurate laser-based techniques. In this manner, it will be possible to<br />
detect minute changes to the separation of the spacecraft caused by passing<br />
gravitational waves.<br />
Application 10 (Ionocraft (“Lifter”)). An ionocraft or ion-propelled aircraft<br />
(a.k.a. “Lifter”) is an electrohydrodynamic (EHD) device which utilizes<br />
an electrical phenomenon known as the Biefeld-Brown effect to produce thrust<br />
in the air, without requiring combustion or moving parts [101].<br />
The basics of such ion air propulsion were established by T. T. Brown in<br />
1928 and were further developed into the ionocraft by Major A. P. de Seversky<br />
(b)
Applications 85<br />
(a)<br />
Figure 3.10: (a) Ionocraft (“Lifter”). (b) Levitating Lifter. [101]<br />
in 1964. It utilizes two basic pieces of equipment in order to take advantage of<br />
the principle that electric current always flows from negative to positive: tall<br />
metal spikes that are installed over an open wire-mesh grid. High negative<br />
voltage is emitted from the spikes toward the positively charged wire grid, just<br />
like the negative and positive poles on an ordinary battery. As the negative<br />
charge leaves the spike arms, it pelts the surrounding air, putting a negative<br />
charge on some of the surrounding air particles. Such negatively charged air<br />
particles (ions) are attracted downward by the positively charged grid. In their<br />
path from the ion emitter to the collector grid, the ions collide with neutral<br />
air molecules - air particles without electric charge. These collisions thrust a<br />
mass of neutral air downward along with the ions. When they reach the grid,<br />
the negatively charged ions are trapped by the positively charged grid but the<br />
neutral air particles that got pushed along flow right through the open grid<br />
mesh, thus producing a downdraft beneath the ionocraft (ionic wind). The<br />
ionocraft rides on this shaft of air, getting its lift just like a helicopter. The<br />
simplest ionocraft is an equilateral triangular configuration (Figure 3.10(a)),<br />
popularly known as a Lifter, which can be constructed from readily available<br />
parts but requires high voltage for its successful operation. The Lifter works<br />
without moving parts, flies silently, uses only electrical energy and is able to<br />
lift its own weight plus an additional payload (Figure 3.10(b)).<br />
Application 11 (Warren Truss). The rigidity of the triangle [152] has been<br />
exploited in bridge design. The Warren truss (1848) [63] consists of longitudinal<br />
members joined only by angled cross-members which form alternately<br />
inverted equilateral triangular shaped spaces along its length (Figure 3.11).<br />
This ensures that no individual strut, beam or tie is subject to bending<br />
or torsional forces, but only to tension or compression. This configuration<br />
(b)
86 Applications<br />
Figure 3.11: Warren Truss [63]<br />
combines strength with economy of materials and can therefore be relatively<br />
light. It is an improvement of the Neville truss which employs a spacing<br />
configuration of isosceles triangles. The first bridge designed in this way was<br />
constructed at London Bridge Station in 1850.<br />
Application 12 (Flammability Diagram). Flammability diagrams [334]<br />
show the regimes of flammability in mixtures of fuel, oxygen and an inert gas<br />
(typically nitrogen).<br />
The flammability diagram for methane appears in Figure 3.12. Prominent<br />
features are the air-line together with its intersections with the flammability<br />
region which determine the upper (UEL=upper explosive limit) and lower<br />
(LEL=lower explosive limit) flammability limits of methane in air. The nose of<br />
the flammability envelope determines the limiting oxygen concentration (LOC)<br />
below which combustion cannot occur.<br />
Application 13 (Goethe’s Color Triangle). In the Goethe Color Triangle<br />
[145], the vertices of an equilateral triangle are labeled with the three primary<br />
pigments, blue (A), yellow (B) and red (C).<br />
The triangle is then further subdivided as in Figure 3.13 with the subdivisions<br />
grouped into primary (I), secondary (II) and tertiary (III) triangles/colors.<br />
The secondary triangle colors represent the mix of the two adjacent<br />
primary triangle colors and the tertiary triangle colors represent the mix<br />
of the adjacent primary color triangle and the non-adjacent secondary triangle<br />
color. Goethe’s color psychology asserted that this triangle was a diagram of<br />
the human mind and he associated each of its colors with a human emotion.<br />
Subregions of the triangle are thus representative of a corresponding emotional<br />
state.
Applications 87<br />
Figure 3.12: Flammability Diagram [334]<br />
I−C<br />
BLUE<br />
I−A<br />
II−A II−B<br />
III−C<br />
III−A<br />
II−C<br />
RED YELLOW<br />
III−B<br />
Figure 3.13: Goethe’s Color Triangle [145]<br />
I−B
88 Applications<br />
Figure 3.14: Maxwell’s Color Triangle [5]<br />
Application 14 (Maxwell’s Color Triangle). The Maxwell Color Triangle<br />
[5] is a ternary diagram of the three additive primary colors of light (red (R),<br />
green (G), blue (B)).<br />
As such, it displays the complete gamut of colors obtainable by mixing two<br />
or three of them together (Figure 3.14). At the center is the equal energy point<br />
representing true white. This triangle shows the quality aspect of psychophysical<br />
color called chromaticity which includes hue and saturation but not the<br />
quantity aspect comprised of the effective amount of light.<br />
Application 15 (USDA Soil Texture Triangle). The Soil Texture Triangle<br />
[72] is a ternary diagram of sand, silt and clay which is used to classify the<br />
texture class of a soil.<br />
The boundaries of the soil texture classes are shown in Figure 3.15. Landscapers<br />
and gardeners may then use this classification to determine appropriate<br />
soil ammendments, such as adding organic matter like compost, to improve<br />
the soil quality.<br />
Application 16 (QFL Diagram). Clastic sedimentary rock is composed of<br />
discrete fragments (clasts) of materials derived from other minerals. Such rock<br />
can be classified using the QFL diagram [79].<br />
This is a ternary diagram comprised of quartz (Q), feldspar (F) and lithic<br />
(sand) fragments (L). The composition and provenance of sandstone is directly<br />
related to its tectonic environment of formation (Figure 3.16).
Applications 89<br />
Figure 3.15: Soil Texture Triangle [72]<br />
Figure 3.16: QFL (Clastic) Diagram [79]
90 Applications<br />
Figure 3.17: De Finetti Diagram (Genetics) [65]<br />
Application 17 (De Finetti Diagram (Genetics)). De Finetti diagrams<br />
[92] are used to display the genotype frequencies of populations where there are<br />
two alleles and the population is diploid.<br />
In Figure 3.17, the curved line represents the Hardy-Weinberg frequency<br />
as a function of p. The diagram may also be extended to demonstrate the<br />
changes that occur in allele frequencies under natural selection.<br />
Figure 3.18: Fundamental Triangle (Game Theory) [312]<br />
Application 18 (Fundamental Triangle (Game Theory)). Von Neumann<br />
and Morgenstern [312] introduced the Fundamental Triangle (Figure<br />
3.18) in their pathbreaking analysis of three-person game theory.<br />
The imputation vector �α = (α1, α2, α3) satisfies<br />
αi ≥ −1 (i = 1, 2, 3); α1 + α2 + α3 = 0.<br />
Thus, the shaded region of Figure 3.18 may be coordinatized as a ternary<br />
diagram which can then be used to determine all solutions for essential zerosum<br />
three-person games. Furthermore, it may be adapted for the analysis of<br />
essential nonzero-sum three-person games.
Applications 91<br />
(a)<br />
Figure 3.19: Representation Triangle: (a) Ranking Regions. (b) Voting Paradox.<br />
[261]<br />
Application 19 (Representation Triangle (Voting Theory)). According<br />
to D. G. Saari [261], each of three candidates may be assigned to the vertex<br />
of an equilateral “representation triangle” which is then subdivided into six<br />
“ranking regions” by its altitudes (Figure 3.19(a)) [261].<br />
Each voter ranks the three candidates. The number of voters of each type<br />
is then placed into the appropriate ranking region (Figure 3.19(b)). The representation<br />
triangle is a useful device to illustrate voting theory paradoxes and<br />
counterintuitive outcomes. The two regions adjacent to a vertex correspond to<br />
first place votes, the two regions adjacent to these second place votes, and the<br />
two most remote regions third place votes. Geometrically, the left-side of the<br />
triangle is closer to A than to B and so represents voters preferring A to B;<br />
ditto for the five regions likewise defined. For the displayed example, C wins a<br />
plurality vote with 42 first place votes. In so-called Borda voting, where a first<br />
place vote earns two points, a second place vote earns one point and a third<br />
place vote earns no points, B wins with a Borda count of 128. In Condorcet<br />
voting, where the victor wins all pairwise elections, A is the Condorcet winner.<br />
So, who really won the election [261]?<br />
Application 20 (Error-Correcting Code). The projective plane of order<br />
2, which is modeled by an equilateral triangle together with its incircle and<br />
altitudes, is shown in Figure 3.20(a).<br />
There are 7 points (numbered) and 7 lines (one of which is curved); each<br />
line contains three points and each point lies on three lines. This is also known<br />
as the Steiner triple-system of order 7 since each pair of points lies on exactly<br />
one line. The matrix representation is shown in Figure 3.20(b) where the rows<br />
(b)
92 Applications<br />
(a)<br />
(b)<br />
Figure 3.20: (a) Projective Plane of Order 2. (b) Binary Matrix Representation.<br />
(c) Hamming Code of Length 7. [286]<br />
represent lines, the columns represent points and the presence of a 1 indicates<br />
that a point lies on a line or, equivalently, a line contains a point (0 otherwise).<br />
The Hamming code of length 7, which contains 8 codewords, is obtained by<br />
taking the complement of the rows of this matrix and appending the zero<br />
codeword (Figure 3.20(c)). It has minimum Hamming distance d = 4 and is a<br />
single-error-correcting code [286].<br />
Application 21 (Equilateral Triangle Rule (Speaker Placement)).<br />
Stereo playback assumes a symmetrical loudspeaker and listener arrangement<br />
with a 60 ◦ angle between the loudspeakers and corresponding to an equilateral<br />
triangular configuration (Figure 3.21) [208].<br />
Application 22 (Equilateral Triangular Microphone Placement). Hioka<br />
and Hamada [173] have explored an algorithm for speaker direction tracking using<br />
microphones located at the vertices of an equilateral triangle (Figure 3.22).<br />
In teleconferencing and remote lecturing systems, speaker direction tracking<br />
is essential for focusing the desired speech signal as well as steering the<br />
camera to point at the speaker. For these applications, the accuracy should be<br />
spatially uniform for omni-directional tracking and, for practical purposes, a<br />
small number of microphones is desirable. Both of these objectives are achieved<br />
by the integrated use of three cross-spectra from the equilateral triangular microphone<br />
array. Computer simulations and experimental measurements have<br />
confirmed that this array possesses uniform omni-directional accuracy and does<br />
not lose track of the speaker even if he/she moves abruptly.<br />
(c)
Applications 93<br />
Figure 3.21: Equilateral Triangle Rule [280]<br />
Figure 3.22: Equilateral Triangular Microphone Array [173]
94 Applications<br />
(a)<br />
Figure 3.23: Loudspeaker Array: (a) Icosahedral Speaker. (b) Equilateral<br />
Triangle Array. [13]<br />
Application 23 (Icosahedral Speaker). The research team at The Center<br />
for New Music and Audio Technologies (CNMAT) of UC-Berkeley, in collaboration<br />
with Meyer Sound of Berkeley, California, has created a compact 120channel<br />
approximately spherical loudspeaker for experiments with synthesis of<br />
acoustic signals with real-time programmable directional properties.<br />
These directional patterns can reproduce the complete radiative signature<br />
of natural instruments or explore new ideas in spatial audio synthesis. A<br />
special hybrid geometry is used that combines the maximal symmetry of a<br />
twenty-triangular-faceted icosahedron (Figure 3.23(a)) with the compact planar<br />
packing of six circles on an equilateral triangle (Figure 3.23(b) shows the<br />
resulting “billiard ball packing”.) [13].<br />
Application 24 (Superconducting Sierpinski Gasket). In 1986, Gordon<br />
et al. [147] reported on their experimental investigations of the properties of a<br />
superconducting Sierpinski gasket (SG) network in a magnetic field.<br />
Because of their dilational symmetry, statistical mechanical and transport<br />
problems are exactly solvable on these fractals. Moreover, study of the SG<br />
network is inherently interesting because of its lack of translational invariance<br />
and its anomalous (fractal) dimensionalities. The experimental gaskets (Figure<br />
3.24) were of tenth order with elementary triangles of area 1.38 µm 2 and<br />
produced excellent quantitative agreement with theoretical predictions.<br />
(b)
Applications 95<br />
Figure 3.24: Superconducting Sierpinski Gasket [147]<br />
Figure 3.25: Square Hole Drill (U. S. Patent 4,074,778) [242]
96 Applications<br />
Application 25 (Square Hole Drill). In 1978, U. S. Patent 4,074,778 was<br />
granted for a “Square Hole Drill” based upon the Reuleaux triangle (Figure<br />
3.25) [242].<br />
The resulting square has slightly rounded corners but achieves approximately<br />
99% of the desired area. It was not the first such drill granted a<br />
patent: the Watts drill received U. S. Patent 1,241,176 in 1917!<br />
Figure 3.26: Wankel Engine [303]<br />
Application 26 (Wankel Engine). The Wankel non-reciprocating engine<br />
(Figure 3.26) is a rotary internal combustion engine which has the shape of a<br />
Reuleaux triangle inscribed in a chamber, rather than the usual piston, cylinder<br />
and mechanical valves [303].<br />
This rotary engine, found in Mazda automobiles, has 40% fewer parts and<br />
thus far less weight. Within the Wankel rotor, three chambers are formed by<br />
the sides of the rotor and the wall of the housing. The shape, size, and position<br />
of the chambers are constantly altered by the rotation of the rotor.<br />
Application 27 (Equilateral Triangular Anemometer). Anemometers<br />
are used to measure either wind speed or air pressure, depending on the style<br />
of anemometer [186].<br />
The most familiar form, the cup anemometer, was invented in 1846 by<br />
J. T. R. Robinson and features four hemispherical cups arranged at 90 ◦ angles.<br />
An anemometer’s ability to measure wind speed is limited by friction along<br />
the axis of rotation and aerodynamic drag from the cups themselves. For<br />
this reason, more accurate anemometers feature only three cups arranged in<br />
an equilateral triangle. Modern ultrasonic anemometers, such as that shown
Applications 97<br />
Figure 3.27: Ultrasonic Anemometer [317]<br />
in Figure 3.27, have no moving parts. Instead, they employ bi-directional<br />
ultrasonic transducers which act as both acoustic transmitters and acoustic<br />
receivers. They work on the principle that, when sound travels against/with<br />
the wind, the total transit time is increased/reduced by an amount dependent<br />
upon the wind speed [317].<br />
Application 28 (Natural Equilateral Triangles). Mother Nature has an<br />
apparent fondness for equilateral triangles which is in evidence in Figure 3.28,<br />
where its manifestation in both (a) the nonliving and (b) the living worlds is<br />
on prominent display [54]. (See Appendix A for many more examples.)<br />
Application 29 (Equilateral Triangular Maps). In 1913, B. J. S. Cahill<br />
of Oakland, California patented his butterfly map which is shown in Figure<br />
3.29(a) [132].<br />
It is obtained by inscribing a regular octahedron in the Earth and then<br />
employing gnomonic projection, i.e. projection from the globe’s center, onto<br />
its eight equilateral triangular faces [132]. However, with the highest face<br />
count (20) amongst regular polyhedra, the icosahedron has long been a favorite<br />
among cartographers. In 1943, distinguished Yale economist Irving<br />
Fisher published his Likeaglobe map which is shown in Figure 3.29(b). It<br />
is the product of gnomonic projection of the world to the twenty equilateral<br />
triangular faces of an inscribed icosahedron [132]. In 1954, R. Buckminster<br />
Fuller patented his Dymaxion Skyocean Projection World Map [113] which is
98 Applications<br />
(a)<br />
Figure 3.28: Natural Equilateral Triangles: (a) High Altitude Snow Crystal.<br />
(b) South American Butterfly Species. [54]<br />
(a)<br />
Figure 3.29: Equilateral Triangular Maps: (a) Cahill’s Butterfly Map (Octahedron).<br />
(b) Fisher’s Likeaglobe Map (Icosahedron). [132]<br />
(b)<br />
(b)
Applications 99<br />
also based upon the icosahedron. It differs from Fisher’s Likeaglobe in having<br />
the North and South poles on opposite faces at points slightly off center.<br />
Figure 3.30: 3-Frequency Geodesic Dome [190]<br />
Application 30 (Geodesic Dome). A geodesic dome is a (portion of a)<br />
spherical shell structure based upon a network of great circles (geodesics) lying<br />
on the surface of a sphere that intersect to form triangular elements [190].<br />
Since the sphere encloses the greatest volume for a given surface area, it<br />
provides for economical design (largest amount of internal space and minimal<br />
heat loss due to decreased outer skin surface) and, because of its triangulated<br />
nature, it is structurally stable. In its simplest manifestation [190], the twenty<br />
triangular faces of an inscribed icosahedron are subdivided into equilateral<br />
triangles by partitioning each edge into n (the “frequency”) segments. The<br />
resulting vertices are then projected onto the surface of the circumscribing<br />
sphere. The projected triangles are no longer congruent but are of two varieties<br />
thereby producing vertices with a valence of either five or six. The<br />
resulting geodesic structure has 12 5-valent vertices, 10(n 2 − 1) 6-valent vertices,<br />
30n 2 edges and 20n 2 faces [152]. See Figure 3.30 for the case n = 3.<br />
The geodesic sphere may now be truncated to produce a dome of the desired<br />
height. Such geodesic domes were popularized in the architecture of R. Buckminster<br />
Fuller [113]. About 1960, biochemists employed electron microscopy<br />
to discover that some viruses have recognizable icosahedral symmetry and look<br />
like tiny geodesic domes [59].
100 Applications<br />
(a)<br />
(b) (c) (d)<br />
Figure 3.31: Sphere Wrapping: (a) Mozartkugel. (b) Petals. (c) Square Wrapping.<br />
(d) Equilateral Triangular Wrapping. [75]<br />
Application 31 (Computational Confectionery (Optimal Wrapping)<br />
[75]). Mozartkugel (“Mozart sphere”) is a fine Austrian confectionery composed<br />
of a sphere with a marzipan core (sugar and almond meal), encased in<br />
nougat or praline cream, and coated with dark chocolate (Figure 3.31(a)).<br />
It was invented in 1890 by Paul Fürst in Salzburg (Mozart’s birthplace)<br />
and about 90 million of them are still made and consumed world-wide each<br />
year. Each spherical treat is individually wrapped in a square of aluminum<br />
foil. In order to minimize the amount of wasted material, it is natural to study<br />
the problem of wrapping a sphere by an unfolded shape which will tile the<br />
plane so as to facilitate cutting the pieces of wrapping material from a large<br />
sheet of foil. E. Demaine et al. [75] have considered this problem and shown<br />
that the substitution of an equilateral triangle for the square wrapper leads to<br />
a savings in material. As shown in Figure 3.31(b), they first cut the surface of<br />
the sphere into a number of congruent petals which are then unfolded onto a<br />
plane. The resulting shape may then be enclosed by a square (Figure 3.31(c))<br />
or an equilateral triangle (Figure 3.31(d)). Their analysis shows that the latter<br />
choice results in a material savings of 0.1%. In addition to the direct savings<br />
in material costs, this also indirectly reduces CO2 emissions, thereby partially<br />
alleviating global warming. Way to go Equilateral Triangle!<br />
Application 32 (Triangular Lower and Upper Bounds). Of all triangles<br />
with a given area A, the equilateral triangle has the smallest principal frequency<br />
Λ and the largest torsional rigidity P. Thus for any triangle, we have the lower<br />
bound 2π<br />
4√ 3· √ A ≤ Λ and the upper bound P ≤ √ 3A 2<br />
15 [243].<br />
The principal frequency Λ is the gravest proper tone of a uniform elastic<br />
membrane uniformly stretched and fixed along the boundary of an equilateral<br />
triangle of area A. It has been rendered a purely geometric quantity by dropping<br />
a factor that depends solely on the physical nature of the membrane. P is<br />
the torsional rigidity of the equilateral triangular cross-section, with area A, of<br />
a uniform and isotropic elastic cylinder twisted around an axis perpendicular
Applications 101<br />
(a)<br />
Figure 3.32: Triangular Lower and Upper Bounds: (a) Vibrating Membrane.<br />
(b) Cylinder Under Torsion. [243]<br />
to the cross-section. The couple resisting such torsion is equal to θµP where<br />
θ is the twist or angle of rotation per unit length and µ is the shear modulus.<br />
So defined, P is a purely geometric quantity depending on the shape and<br />
size of the cross-section. Figure 3.32(a) shows one of the vibrational modes<br />
of a triangular membrane while Figure 3.32(b) shows the shear stress in the<br />
cross-section of an equilateral triangular prism under torsion.<br />
T<br />
3<br />
2.5<br />
2<br />
1.5<br />
1<br />
0.5<br />
0<br />
1<br />
0.8<br />
0.6<br />
y<br />
0.4<br />
0.2<br />
0<br />
0<br />
Figure 3.33: Fundamental Mode [219]<br />
Application 33 (Laplacian Eigenstructure). The eigenvalues and eigenfunctions<br />
of the Laplace operator, ∆ := ∂2<br />
∂x2 + ∂2<br />
∂y2, on the equilateral triangle<br />
play an important role in Applied Mathematics.<br />
0.2<br />
0.4<br />
x<br />
0.6<br />
0.8<br />
1<br />
(b)
102 Applications<br />
The eigenstructure of the Laplacian arises in heat transfer, vibration theory,<br />
acoustics, electromagnetics and quantum mechanics to name but a few of its<br />
ubiquitous appearances in science and engineering. G. Lamé discovered explicit<br />
formulas for the cases of Dirichlet and Neumann boundary conditions which<br />
were later extended to the Robin boundary condition by B. J. McCartin [219].<br />
Figure 3.33 shows the fundamental mode for the Dirichlet boundary condition.
Chapter 4<br />
Mathematical Recreations<br />
The impact of the equilateral triangle is considerably extended if we broaden<br />
our scope to include the rich field of Recreational Mathematics [273].<br />
Figure 4.1: Greek Symbol Puzzle [210]<br />
Recreation 1 (Sam Loyd’s Greek Symbol Puzzle [210]). Draw the Greek<br />
symbol of Figure 4.1 in one continuous line making the fewest possible number<br />
of turns (going over the same line as often as one wishes).<br />
The displayed solution commences at A and terminates at B with segment<br />
AB traced twice. It requires only 13 turns (14 strokes). If no segment is traced<br />
twice then 14 turns are necessary and sufficient. This same puzzle appeared<br />
in [87].<br />
103
104 Mathematical Recreations<br />
Figure 4.2: Dissected Triangle [86]<br />
Recreation 2 (H. E. Dudeney’s Dissected Triangle [86]). Cut a paper<br />
equilateral triangle into five pieces in such a way that they will fit together and<br />
form either two or three smaller equilateral triangles, using all the material in<br />
each case.<br />
In Figure 4.2, diagram A is the original triangle, which for the sake of<br />
definiteness is assumed to have an edge of 5 units in length, dissected as shown.<br />
For the two-triangle solution, we have region 1 together with regions 2, 3, 4 and<br />
5 assembled as in diagram B. For the three-triangle solution, we have region<br />
1 together with regions 4 and 5 assembled as in diagram C and regions 2 and<br />
3 assembled as in diagram D. Observe that in diagrams B and C, piece 5 has<br />
been turned over which was not prohibited by the statement of the problem.<br />
(a) (b)<br />
Figure 4.3: (a) Triangle Dissection. (b) Triangle Hinged Dissection. [85]<br />
Recreation 3 (H. E. Dudeney’s Haberdasher’s Puzzle [85]). The Haberdasher’s<br />
Puzzle [85] concerns cutting an equilateral triangular piece of cloth<br />
into four pieces that can be rearranged to make a square.
Mathematical Recreations 105<br />
With reference to Figure 4.3(a): Bisect AB in D and BC in E; produce the<br />
line AE to F making EF equal to EB; bisect AF in G and describe arc AHF;<br />
produce EB to H, and EH is the length of the side of the required square; from<br />
E with distance EH, describe the arc HJ, and make JK equal to BE; now from<br />
the points D and K drop perpendiculars on EJ at L and M. The four resulting<br />
numbered pieces may be reassembled to form a square as in the Figure. Note<br />
that AD, DB, BE, JK are all equal to half the side of the triangle [97]. Also,<br />
LJ=ME [66]. As shown in Figure 4.3(b), the four pieces can be hinged in such a<br />
way that the resulting chain can be folded into either the square or the original<br />
triangle. Dudeney himself displayed such a table made of polished mahogany<br />
and brass hinges at the Royal Society in 1905 [85]. An incorrect version of this<br />
dissection appeared as Steinhaus’ Mathematical Snapshot #2 where the base<br />
is divided in the ratio 1:2:1 (the correct ratios are approximately 0.982:2:1.018)<br />
[291]. This was corrected as Schoenberg’s Mathematical Time Exposure #1<br />
[272] (independently of Dudeney).<br />
Figure 4.4: Triangle and Square Puzzle [87]<br />
Recreation 4 (H. E. Dudeney’s Triangle and Square Puzzle [87]). It<br />
is required to cut each of two equilateral triangles into three pieces so that the<br />
six pieces fit together to form a perfect square [87].<br />
Cut one triangle in half and place the pieces together as in Figure 4.4(1).<br />
Now cut along the dotted lines, making ab and cd each equal to the side of the<br />
required square. Then, fit together the six pieces as in Figure 4.4(2), sliding<br />
the pieces F and C upwards and to the left and bringing down the little piece<br />
D from one corner to the other.<br />
Recreation 5 (H. E. Dudeney’s Square and Triangle Puzzle [87]). It<br />
is required to fold a perfectly square piece of paper so as to form the largest<br />
possible equilateral triangle [87]. (See Property 2.21.)
106 Mathematical Recreations<br />
Figure 4.5: Square and Triangle Puzzle [87]<br />
With reference to Figure 4.5, fold the square in half and make the crease<br />
FE. Fold the side AB so that the point B lies on FE, and you will get the<br />
points F an H from which you can fold HGJ. While B is on G, fold AB back<br />
on AH, and you will have the line AK. You can now fold the triangle AJK,<br />
which is the largest possible equilateral triangle obtainable.<br />
Figure 4.6: Triangle-to-Triangle Dissection [214]<br />
Recreation 6 (Triangle-to-Triangle Dissection [214]). A given equilateral<br />
triangle can be dissected into noncongruent pieces that can be rearranged<br />
to produce the original triangle in two different ways.<br />
Such a dissection of an equilateral triangle into eight pieces is shown in<br />
Figure 4.6 [214]. This interesting dissection was found by superimposing two<br />
different strips of triangular elements. Incidentally, this is not a minimal-piece<br />
dissection.<br />
Recreation 7 (Polygonal Dissections [109]). Mathematicians’ appetite for<br />
polygonal dissections never seems to be sated [109].
Mathematical Recreations 107<br />
(a) (b) (c)<br />
Figure 4.7: Polygon-to-Triangle Dissections; (a) Pentagon. (b) Hexagon. (c)<br />
Nonagon (Enneagon). [109]<br />
Witness Figure 4.7: (a) displays Goldberg’s six piece dissection of a regular<br />
pentagon; (b) displays Lindgren’s five piece dissection of a regular hexagon; (c)<br />
displays Theobald’s eight piece dissection of a regular nonagon (enneagon). All<br />
three have been reassembled to form an equilateral triangle and all are believed<br />
to be minimal dissections [109].<br />
Figure 4.8: Dissections into Five Isosceles Triangles [135]<br />
Recreation 8 (Dissection into Five Isosceles Triangles [135]). Figure<br />
4.8 shows four ways to cut an equilateral triangle into five isosceles triangles<br />
[135].<br />
The four patterns, devised by R. S. Johnson, include one example of no<br />
equilateral triangles among the five, two examples of one equilateral triangle<br />
and one example of two equilateral triangles. H. L. Nelson has shown that<br />
there cannot be more than two equilateral triangles.
108 Mathematical Recreations<br />
(a) (b) (c)<br />
Figure 4.9: Three Similar Pieces: (a) All Congruent. (b) Two Congruent. (c)<br />
None Congruent. [138]<br />
Recreation 9 (Dissection into Three Parts [138]). It is easy to trisect<br />
an equilateral triangle into three congruent pieces as in Figure 4.9(a).<br />
It is much more difficult to dissect it into three similar parts, just two of<br />
which are congruent as has been done in Figure 4.9(b). Yet, to dissect the<br />
triangle into three similar pieces, none of which are congruent, is again easy<br />
(see Figure 4.9(c)) [138].<br />
Figure 4.10: Trihexaflexagon [121]<br />
Recreation 10 (Trihexaflexagon [121]). Flexagons are paper polygons which<br />
have a surprising number of faces when “flexed” [121].
Mathematical Recreations 109<br />
To form a trihexaflexagon, begin with a strip of paper with ten equilateral<br />
triangles numbered as shown in Figure 4.10. Then, fold along ab, fold along cd,<br />
fold back the protruding triangle and glue it to the back of the adjacent triangle<br />
and Voila! The assembled trihexaflexagon is a continuous band of hinged triangles<br />
with a hexagonal outline (“face”). If the trihexaflexagon is “pinch-flexed”<br />
[244], as shown, then one face will become hidden and a new face appears.This<br />
remarkable geometrical construction was discovered by Arthur H. Stone when<br />
he was a Mathematics graduate student at Princeton University in 1939. A<br />
Flexagon Committe consisting of Stone, Bryant Tuckerman, Richard P. Feynman<br />
and John W. Tukey was formed to probe its mathematical properties<br />
which are many and sundry [244].<br />
Figure 4.11: Bertrand’s Paradox [122]<br />
Recreation 11 (Bertrand’s Paradox [122]). The probability that a chord<br />
drawn at random inside a circle will be longer than the side of the inscribed<br />
equilateral triangle is equal to 1 1 , 3 2<br />
and 1<br />
4 [122]!<br />
With reference to the top of Figure 4.11, if one endpoint of the chord is fixed<br />
at A and the other endpoint is allowed to vary then the probability is equal<br />
to 1.<br />
Alternatively (Figure 4.11 bottom left), if the diameter perpendicular to<br />
3<br />
the chord is fixed and the chord allowed to slide along it then the probability<br />
is equal to 1.<br />
Finally (Figure 4.11 bottom right), if both endpoints of the<br />
2<br />
chord are free and we focus on its midpoint then the required probability is<br />
computed to be 1.<br />
Physical realizations of all three scenarios are provided in<br />
4<br />
[122] thus showing that caution must be used when the phrase “at random” is<br />
bandied about, especially in a geometric context.
110 Mathematical Recreations<br />
Figure 4.12: Two-Color Map [123]<br />
Recreation 12 (Two-Color Map [123]). How can a two-color planar map<br />
be drawn so that no matter how an equilateral triangle with unit side is placed<br />
on it, all three vertices never lie on points of the same color [123]?<br />
A simple solution is shown in Figure 4.12 where the vertical stripes are<br />
closed on the left and open on the right [123]. It is an open problem as to how<br />
many colors are required so that no two points, a unit distance apart, lie on<br />
the same color. However, it is known that four colors are necessary and seven<br />
colors are sufficient.<br />
(a) (b)<br />
Figure 4.13: Sphere Coloring: (a) The Problem. (b) Five Colors Are Sufficient.<br />
[131]
Mathematical Recreations 111<br />
Recreation 13 (Erdös’ Sphere Coloring Problem [131]). Paul Erdös<br />
proposed the following unsolved problem in graph theory. What is the minimum<br />
number of colors required to paint all of the points on the surface of a unit<br />
sphere so that, no matter how we inscribe an equilateral triangle of side √ 3<br />
(the largest such triangle that can be so inscribed), the triangle will have each<br />
corner on a different color (Figure 4.13(a)) [131]?<br />
E. G. Straus has shown that five colors suffice. In his five-coloring (shown in<br />
Figure 4.13(b)), the north polar region is open with boundary circle of diameter<br />
√ 3. The rest of the sphere is divided into four identical regions, each closed<br />
along its northern and eastern borders, as indicated by the heavy black line on<br />
the dark shaded region. One color is given to the cap and to the south pole.<br />
The remaining four colors are assigned to four quadrant regions. G. J. Simmons<br />
has shown that three colors are not sufficient so that at least four colors are<br />
necessary. It is unknown whether four or five colors are both necessary and<br />
sufficient. The analogous problem for the plane, i.e. the minimum number of<br />
colors which ensures that every equilateral triangle of unit side will have its<br />
corners on different colors, is also open. Indeed, it is equivalent to asking for a<br />
minimal coloring of the plane so that every unit line segment has its endpoints<br />
on different colors, a problem which was discussed at the end of the previous<br />
Recreation. This problem may be recast in terms of the chromatic number of<br />
planar graphs [131].<br />
(a) (b) (c)<br />
Figure 4.14: Optimal Spacing of Lunar Bases: (a) n=3. (b) n=4. (c) n=12.<br />
[124]<br />
Recreation 14 (Optimal Spacing of Lunar Bases [124]). Assume that<br />
the moon is a perfect sphere and that we want to establish n lunar bases as far<br />
apart from one another as possible.<br />
I.e., how can n points be arranged on a sphere so that the smallest distance<br />
between any pair of points is maximized? This problem is equivalent to that
112 Mathematical Recreations<br />
of placing n equal, nonoverlapping circles on a sphere so that the radius of<br />
each circle is maximized [124]. The solution for the cases n = 3, 4, 12 appear<br />
in Figure 4.14 and all involve equilateral triangles. The solution for cases<br />
2 ≤ n ≤ 12 and n = 24 are known, otherwise the solution is unknown [124].<br />
Figure 4.15: Rep-4 Pentagon: The Sphinx [146]<br />
Recreation 15 (Replicating Figures: Rep-tiles [125]). In 1964, S. W.<br />
Golomb gave the name “rep-tile” to a replicating figure that can be used to assemble<br />
a larger copy of itself or, alternatively, that can be dissected into smaller<br />
replicas of itself [146]. If four copies are required then this is abbreviated rep-4.<br />
Figure 4.15 contains a rep-4 pentagon, known as the Sphinx, which may be<br />
regarded as composed of six equilateral triangles or two-thirds of an equilateral<br />
triangle [146]. The Sphinx is the only known 5-sided rep-tile [296, p. 134].<br />
Figure 4.16 contains three examples of rep-4 nonpolygonal figures composed<br />
of equilateral triangles: the Snail, the Lamp and the Carpenter’s Plane [125].<br />
Each of these figures, shown at the left, is formed by adding to an equilateral<br />
triangle an endless sequence of smaller triangles, each one one-fourth the size<br />
of its predecessor. In each case, four of these figures will fit together to make a<br />
larger replica, as shown on the right. (There is a gap in each replica because the<br />
original figure cannot be drawn with an infinitely long sequence of triangles.)<br />
Recreation 16 (Hexiamonds [126]). Hexiamonds were invented by S. W.<br />
Golomb in 1954 and officially named by T. H. O’Beirne in 1961. Each hexiamond<br />
is composed of six equilateral triangles joined along their edges. Treating<br />
mirror images as identical, there are exactly 12 of them (Figure 4.17) [126].<br />
Much is known about the mathematical properties of hexiamonds. For<br />
example, the six-pointed star of Figure 4.18(a) is known to have the unique<br />
eight-piece solution of Figure 4.18(b) [126]. Sets of plastic hexiamonds were<br />
marketed in the late 1960’s, under various trade names, in England, Japan,<br />
and West Germany.
Mathematical Recreations 113<br />
Figure 4.16: Rep-4 Non-Polygons: The Snail, The Lamp, and The Carpenter’s<br />
Plane [125]<br />
Recreation 17 (MacMahon’s 24 Color Triangles [128]). In 1921, Major<br />
Percy A. MacMahon, a noted combinatorialist, introduced a set of 24 color<br />
triangles [213], the edges of which are colored with one of four colors, that<br />
are pictured in Figure 4.19 [128]. (Rotations of triangles are not considered<br />
different but mirror-image pairs are considered distinct.) The pieces are to<br />
be fitted together with adjacent edges matching in color to form symmetrical<br />
polygons, the border of which must all be of the same color.<br />
It is known that all polygons so assembled from the 24 color triangles<br />
must have perimeters of 12, 14, or 16 unit edges. Also, only one polygon, the<br />
regular hexagon, has the minimum perimeter of 12. Its one-color border can<br />
be formed in six different ways, each with an unknown number of solutions.<br />
For each type of border, the hexagon can be solved with the three triangles of<br />
solid color (necessarily differing in color from the border) placed symmetrically<br />
around the center of the hexagon. Since each solid-color triangle must be<br />
surrounded by triangular segments of the same color, the result is three smaller<br />
regular hexagons of solid color situated symmetrically at the center of the larger<br />
hexagon. Figure 4.20 displays a hexagon solution for each of the six possible<br />
border patterns [128]. As previously noted, it is not known how many solutions<br />
there are of these six types although the total number of solutions has been<br />
estimated to be in the thousands.
114 Mathematical Recreations<br />
Figure 4.17: The 12 Hexiamonds [126]
Mathematical Recreations 115<br />
(a)<br />
Figure 4.18: Hexiamond Star: (a) Problem. (b) Solution. [126]<br />
(b)<br />
Figure 4.19: MacMahon’s 24 Color Triangles [128]
116 Mathematical Recreations<br />
Figure 4.20: Six Solutions to the Hexagon Problem [128]
Mathematical Recreations 117<br />
(a)<br />
Figure 4.21: Icosahedron: (a) Icosahedron Net. (b) Five-Banded Icosahedron.<br />
[130]<br />
Recreation 18 (Plaited Polyhedra [130]). Traditionally [66], paper models<br />
of the five Platonic solids are constructed from “nets” like that shown in Figure<br />
4.21(a) for the icosahedron. The net is cut out along the solid line, folded<br />
along the dotted lines, and the adjacent faces are then taped together. In 1973,<br />
Jean J. Pedersen of Santa Clara University discovered a method of weaving or<br />
braiding (“plaiting”) the Platonic solids from n congruent straight strips. Each<br />
strip is of a different color and each model has the properties that every edge<br />
is crossed at least once by a strip, i.e. no edge is an open slot, and every color<br />
has an equal area exposed on the model’s surface. (An equal number of faces<br />
will be the same color on all Platonic solids except the dodecahedron, which has<br />
bicolored faces when braided by this technique.) She has proved that if these<br />
two properties are satisfied then the number of necessary and sufficient bands<br />
for the tetrahedron, cube, octahedron, icosahedron and dodecahedron are two,<br />
three, four, five and six, respectively [130].<br />
With reference to Figure 4.21(b), the icosahedron is woven with five valleycreased<br />
strips. A visually appealing model can be constructed with each color<br />
on two pairs of adjacent faces, the pairs diametrically opposite each other.<br />
All five colors go in one direction around one corner and in the opposite direction,<br />
in the same order, around the diametrically opposite corner. Each<br />
band circles an “equator” of the icosahedron, its two end triangles closing the<br />
band by overlapping. In making the model, when the five overlapping pairs of<br />
ends surround a corner, all except the last pair can be held with paper clips,<br />
which are later removed. The last overlapping end then slides into the proper<br />
slot. Experts may dispense with the paper clips [130]. Previous techniques of<br />
polyhedral plaiting involved nets of serpentine shape [322].<br />
Recreation 19 (Pool-Ball Triangles [133]). Colonel George Sicherman of<br />
Buffalo asked while watching a game of pool: Is it possible to form a “difference<br />
triangle” in arranging the fifteen balls in the usual equilateral triangular<br />
configuration at the beginning of a game?<br />
(b)
118 Mathematical Recreations<br />
Figure 4.22: Pool-Ball Triangle [133]<br />
In a difference triangle, the consecutive numbers are arranged so that each<br />
number below a pair of numbers is the positive difference between that pair.<br />
He easily found two solutions for three balls and four solutions each for six and<br />
ten balls. However, he was surprised to discover that, for all fifteen balls, there<br />
is only the single solution shown in Figure 4.22, up to reflection. Incidentally,<br />
it has been proved that no difference triangle can have six or more rows [295,<br />
p. 7].<br />
(a) (b)<br />
Figure 4.23: Equilateral Triangular Billiards: (a) Triangular Pool Table. (b)<br />
Unfolding Billiard Orbits. [198]<br />
Recreation 20 (Equilateral Triangular Billiards [198]). In the “billiard<br />
ball problem”, one seeks periodic motions of a billiard ball on a convex billiard<br />
table, where the law of reflection at the boundary is that the angle of incidence<br />
equals the angle of reflection [24, pp. 169-179]. Even for triangular pool tables,<br />
the present state of our knowledge is very incomplete. For example, it is not<br />
known if every obtuse triangle possesses a periodic orbit and, for a general<br />
non-equilateral acute triangle, the only known periodic orbit is the Fagnano<br />
orbit consisting of the pedal triangle (see Figure 2.35 right) [158]. However,<br />
the equilateral triangular billiard table possesses infinitely many periodic orbits<br />
[16].
Mathematical Recreations 119<br />
(a)<br />
(b)<br />
Figure 4.24: Triangular Billiards Redux: (a) Notation. (b) Some Orbits. (c)<br />
Gardner’s Gaffe. [198]<br />
In his September 1963 “Mathematical Games” column in Scientific American<br />
[118], Martin Gardner put forth a flawed analysis of periodic orbits on<br />
an equilateral triangular billiards table. In 1964, this motivated D. E. Knuth<br />
to provide a correct, simple and comprehensive analysis of periodic billiard<br />
orbits on an equilateral triangular table (see Figure 4.23(a)) [198]. His analysis<br />
involves “unfolding” periodic orbits using the Schwarz reflection procedure<br />
previously alluded to in our prior treatment of Fagnano’s problem of the shortest<br />
inscribed triangle in Chapter 2 (Property 53). This results in the tiling of<br />
the plane by repeated reflections of the billiard table ABC shown in Figure<br />
4.23(b). Straight lines superimposed on this figure, such as L,M,N,P, give<br />
paths in the original triangle satisfying the reflection law if the diagram is<br />
folded appropriately. Conversely, any infinite path satisfying the reflection law<br />
corresponds to a straight line in this diagram.<br />
Turning our attention to Figure 4.24(a) where AB has unit length, 0 <<br />
x < 1 is the point from which the billiard ball is launched at an angle θ and we<br />
wish to determine those launching angles which result in endlessly repeating<br />
periodic trajectories. The particular trajectory shown there corresponds to the<br />
important special case θ = 60 ◦ and is associated with path M of Figure 4.23(b).<br />
If x = 1/3 then this would coincide with the closed light path of Figure 2.35.<br />
If x = 1/2 then this becomes the Fagnano orbit and we (usually) exclude<br />
this highly specialized orbit from further consideration in what follows. Two<br />
other special cases deserve attention: θ = 90 ◦ (Figure 4.24(b)left and path L<br />
in Figure 4.23(b)) and θ = 30 ◦ (Figure 4.24(b) right and path P in Figure<br />
4.23(b)), which are unusual in that one half of the path retraces the other half<br />
in the opposite direction. A generic periodic orbit is shown in Figure 4.24(b)<br />
center which corresponds to path N in Figure 4.23(b).<br />
Now that the Fagnano orbit has been effectively excluded, Knuth shows<br />
that periodic orbits correspond to lines in Figure 4.23(b) connecting the original<br />
x to one of its images on a horizontal line and these are labeled with<br />
coordinates (i, j) of two types, either (m, n) or (m + 1/2, n + 1/2), where m<br />
(c)
120 Mathematical Recreations<br />
and n are integers. He then states and proves his main results:<br />
Theorem 4.1. A path is cyclic if and only if θ = 90 ◦ , or if tanθ = r/ √ 3,<br />
where r is a nonzero rational number. Moreover,<br />
• The length of the path traveled in each cycle may be determined as follows:<br />
Let tanθ = p/(q √ 3) where p and q are integers with no common factor,<br />
and where p > 0, q ≥ 0. Then the length is k � 3p 2 + 9q 2 where k = 1/2<br />
if p and q are both odd, k = 1 otherwise; except when θ = 60 ◦ and<br />
x = 1/2 (Fagnano orbit), when the length is 3/2.<br />
• If x �= 1/2, the shortest path length is √ 3 and it occurs when θ = 30 ◦<br />
and θ = 90 ◦ .<br />
• The number of bounces occurring in each cycle may be determined as follows<br />
(when the path leads into a corner, this is counted as three bounces,<br />
as can be justified by a limiting argument): With p, q, k as above, the<br />
number of bounces is k[2p + min (2p, 6q)]; except when θ = 60 ◦ and<br />
x = 1/2 (Fagnano orbit), when the number of bounces is 3.<br />
• If x �= 1/2, the least number of bounces per cycle is 4 and it occurs when<br />
θ = 30 ◦ and θ = 90 ◦ . With the exception of the Fagnano orbit, the<br />
number of bounces is always even.<br />
Finally, Knuth takes up Gardner’s gaffe where he claimed that the path<br />
obtained by folding Figure 4.24(c) yields a periodic orbit. However, unless<br />
θ = 60 ◦ and x = 1/2, the angle of incidence is not equal to the angle of<br />
reflection at x. Knuth then shows that the extended path is cyclical if and only<br />
if x is rational. Baxter and Umble [16], evidently unaware of Knuth’s earlier<br />
pathbreaking work on equilateral triangular billiards, provide an analysis of<br />
this problem ab initio. However, they introduce an equivalence relation on<br />
the set of all periodic orbits where equivalent periodic orbits share the same<br />
number of bounces, path length and incidence angles (up to permutation).<br />
They also count the number of equivalence classes of orbits with a specified<br />
(even) number of bounces.<br />
Recreation 21 (Tartaglian Measuring Puzzles [62]). The following liquidpouring<br />
puzzle is due to the Renaissance Mathematician Niccolò Fontana,<br />
a.k.a. Tartaglia (“The Stammerer”) [118]. An eight-pint vessel is filled with<br />
water. By means of two empty vessels that hold five and three pints respectively,<br />
divide the eight pints evenly between the two larger vessels by pouring<br />
water from one vessel into another. In any valid solution, you are not allowed<br />
to estimate quantities, so that you can only stop pouring when one of the vessels<br />
becomes either full or empty. In 1939, M. C. K. Tweedie [311] showed<br />
how to solve this and more general pouring problems by utilizing the trajectory<br />
of a bouncing ball upon an equilateral triangular lattice (Figure 4.25).
Mathematical Recreations 121<br />
Figure 4.25: Tartaglian Measuring Puzzle [62]<br />
In the ternary diagrams of Figure 4.25, the horizontal lines correspond to<br />
the contents of the the 8-pint vessel while the downward/upward slanting lines<br />
correspond to that of the 5/3-pint vessel, respectively. The closed highlighted<br />
parallelogram corresponds to the possible states of the vessels with those on<br />
the boundary corresponding to the states with one or more of the vessels either<br />
completely full or completely empty, i.e. the valid intermediate states in any<br />
proposed solution. The number triplets indicate how much each vessel holds<br />
at any stage of the solution process in the order (8-pint, 5-pint, 3-pint).<br />
Starting at the apex marked by the initial state 800, the first move must<br />
be to fill either the 5-pint vessel (Figure 4.25 left) or the 3-pint vessel (Figure<br />
4.25 right). Thereafter, we follow the path of a billiard ball bouncing on the<br />
indicated parallelogram until finally reaching the final state 440. The law of<br />
reflection is justified by the fact that each piece of the broken lines, shown<br />
hashed in Figure 4.25, are parallel to a side of the outer triangle of reference<br />
and so represent the act of pouring liquid from one vessel into another while<br />
the third remains untouched. Figure 4.25 left thereby yields the seven-step<br />
solution:<br />
800 → 350 → 323 → 620 → 602 → 152 → 143 → 440,<br />
while Figure 4.25 right generates the eight-step solution:<br />
800 → 503 → 530 → 233 → 251 → 701 → 710 → 413 → 440.<br />
A more detailed study of this technique, especially as to its generalizations<br />
and limitations, is available in the literature [118, 229, 296]<br />
Recreation 22 (Barrel Sharing [284]). Barrel sharing problems have been<br />
common recreational problems since at least the Middle Ages [284]. In their<br />
simplest manifestation, N full, N half-full and N empty barrels are to be shared
122 Mathematical Recreations<br />
Figure 4.26: Barrel Sharing (N = 5) [284]<br />
among three persons so that each receives the same amount of contents and the<br />
same number of barrels. Using equilateral triangular coordinates, D. Singmaster<br />
[284] has shown that the solutions of this problem correspond to triangles<br />
with integer sides and perimeter N.<br />
Suppose that there are N barrels of each type (full, half-full and empty)<br />
and let fi, hi, ei be the nonnegative integer number of these that the ith<br />
person receives (i = 1, 2, 3). Then, a fair sharing is defined as one satisfying<br />
the following conditions:<br />
fi + hi + ei = N, fi + hi<br />
2<br />
+ N<br />
2<br />
(i = 1, 2, 3);<br />
3�<br />
fi =<br />
In turn, these conditions lead to the equivalent conditions:<br />
ei = fi, hi = N − 2fi, fi ≤ N<br />
2<br />
i=1<br />
(i = 1, 2, 3);<br />
3�<br />
hi =<br />
i=1<br />
3�<br />
ei = N.<br />
i=1<br />
3�<br />
fi = N.<br />
However, three nonnegative lengths x, y, z can form a triangle if and only<br />
if the three triangle inequalities hold:<br />
x + y ≥ z, y + z ≥ x, z + x ≥ y.<br />
Setting x + y + z = p, this is equivalent to<br />
x ≤ p p p<br />
, y ≤ , z ≤<br />
2 2 2 .<br />
i=1
Mathematical Recreations 123<br />
Hence, the solutions for sharing N barrels of each type are just the integral<br />
lengths that form a triangle of perimeter N. Consider a triangle of sides x, y,<br />
z and perimeter p. Since x + y + z = p, we can establish the ternary diagram<br />
of Figure 4.26 (N = 5) where the dashed central region corresponds to the<br />
inequalities: x ≤ p/2, y ≤ p/2, z ≤ p/2. The three (equivalent) solutions<br />
contained in this region correspond to two persons receiving 2 full, 1 half-full<br />
and 2 empty barrels, and one person receiving 1 full, 3 half-full and 1 empty<br />
barrel. Singmaster [284] includes in his analyis a counting procedure for the<br />
number of non-equivalent solutions and considers more general barrel sharing<br />
problems.<br />
Figure 4.27: Fraenkel’s “Traffic Jam” Game [135]<br />
Recreation 23 (“Traffic Jam” Game [135]). A. S. Fraenkel, a mathematician<br />
at the Weizmann Institute of Science in Israel, invented the game “Traffic<br />
Jam” which is played on the directed graph shown in Figure 4.27 [135].<br />
A coin is placed on each of the four shaded spots A, D, F and M. Players<br />
take turns moving any one of the coins along one of the directed edges of the<br />
graph to an adjacent spot whether or not that spot is occupied. (Each spot can<br />
hold any number of coins.) Note that A is a source (all arrows point outwards)<br />
and C is a sink (all arrows point inwards). When all four coins are on sink<br />
C, the person whose turn it is to move has nowhere to move and so loses the<br />
game. J. H. Conway has proved that the first player can always win if and<br />
only if his first move is to from M to L. Otherwise, his opponent can force a<br />
win or a draw, assuming that both players always make their best moves.
124 Mathematical Recreations<br />
(a)<br />
Figure 4.28: Eternity Puzzle: (a) Sample Piece. (b) Puzzle Board. [313]<br />
(b)<br />
(a) (b)<br />
Figure 4.29: Eternity Puzzle Solutions: (a) Selby-Riordan. (b) Stertenbrink.<br />
[238]
Mathematical Recreations 125<br />
Recreation 24 (Eternity Puzzle [242]). The Eternity Puzzle [242] is a<br />
jigsaw puzzle comprised of 209 pieces constructed from 12 hemi-equilateral<br />
(30 ◦ − 60 ◦ − 90 ◦ ) triangles (Figure 4.28(a)). These pieces must be assembled<br />
into an almost-regular dodecagon on a game board with a triangular grid<br />
(Figure 4.28(b)).<br />
In June 1999, the inventor of the puzzle, Christopher Monckton, offered a<br />
£1M prize for its solution. In May 2000, two mathematicians, Alex Selby and<br />
Oliver Riordan, claimed the prize with their solution shown in Figure 4.29(a).<br />
In July 2000, Günter Stertenbrink presented the independent solution shown<br />
in Figure 4.29(b). As these two solutions do not conform to the six clues<br />
provided by Monckton, his solution, which remains unknown, is presumably<br />
different. This is not surprising since it is estimated that the Eternity Puzzle<br />
has on the order of 10 95 solutions (it is estimated that there are approximately<br />
8×10 80 atoms in the observable universe), but these are the only two (three?)<br />
that have been found!<br />
Figure 4.30: Knight’s Tours on a Triangular Honeycomb [316]<br />
Recreation 25 (Knight’s Tours on a Triangular Honeycomb [316]).<br />
The traditional 8 × 8 square chessboard may be replaced by using hexagons<br />
rather than squares and build chessboards, called triangular honeycombs by<br />
their inventor Heiko Harborth of the Technical University of Braunschweig, in<br />
the shape of equilateral triangles. Knight’s Tours for boards of orders 8 and 9<br />
are on display in Figure 4.30 [316].<br />
The subject of Knight’s Tours on the traditional chessboard have a rich<br />
mathematical history [242]. The earliest recorded solution was provided by de<br />
Moivre which was subsequently “improved” by Legendre. Euler was the first<br />
to write a mathematical paper analyzing Knight’s Tours.
126 Mathematical Recreations<br />
Figure 4.31: Nonattacking Rooks on a Triangular Honeycomb [136]<br />
Recreation 26 (Nonattacking Rooks on a Triangular Honeycomb<br />
[136]). The maximum number of nonattacking rooks that can be placed on<br />
a triangular honeycomb of order n in known for 1 ≤ n ≤ 13: 1, 1, 2, 3, 3, 4,<br />
5, 5, 6, 7, 7, 8, 9. Figure 4.31 shows such a configuration of 5 rooks on an<br />
order-8 board [136].<br />
Figure 4.32: Sangaku Geometry [112]<br />
Recreation 27 (Sangaku Geometry [112]). Figure 4.32 portrays a Sangaku<br />
Geometry (“Japanese Temple Geometry”) problem: Express the radius,<br />
c, of the small white circles in terms of the radius, r, of the dashed circle. The<br />
solution is c = r/10 [112, p. 124].<br />
During Japan’s period of isolation from the West (roughly mid-Seventeenth<br />
to mid-Nineteenth Centuries A.D.) imposed by decree of the shogun [242],<br />
Sangaku arose which were colored puzzles in Euclidean geometry on wooden<br />
tablets that were hung under the roofs of Shinto temples and shrines.
Mathematical Recreations 127<br />
Figure 4.33: Paper Folding [258]<br />
Recreation 28 (Paper Folding [258]). Paper folding may be used as a<br />
pedagogical device to expose even preschoolers to elementary concepts of plane<br />
Euclidean geometry [258, p. 13]. Figure 4.33 displays an equilateral triangle<br />
so obtained, replete with altitudes, center and pedal triangle.<br />
Origami, the ancient Japanese art of paper folding, has traditionally focused<br />
on forming the shape of natural objects such as animals, birds and fish<br />
rather than polygons. Nonetheless, it has long held a particular fascination for<br />
many mathematicians such as Lewis Carroll (C. L. Dodgson) [120].<br />
(a) (b) (c)<br />
Figure 4.34: Spidrons: (a) Seahorse. (b) Tiling. (c) Polyhedron. [95]
128 Mathematical Recreations<br />
Recreation 29 (Spidrons [242]). To create a spidron, subdivide an equilateral<br />
triangle by connecting its center to its vertices and reflect one of the<br />
three resulting 30 ◦ − 30 ◦ − 120 ◦ isosceles triangles (with area one-third of the<br />
original equilateral triangle) about one of its shorter sides. Now, construct an<br />
equilateral triangle on one of the shorter sides (also with area one-third of the<br />
original equilateral triangle) and repeat the process of subdivision-reflectionconstruction.<br />
This will produce a spiraling structure with increasingly small<br />
components. By deleting the original triangle we arrive at a semi-spidron and<br />
joining two of them together results in a spidron which is the seahorse shape<br />
of Figure 4.34(a).<br />
Note that, since 2 2 2 + + + · · · = 2 · = 1, the sum of the areas<br />
3 9 27 1−1/3<br />
of the sequence of triangles following an equilateral triangle in a spidron is<br />
equal to the area of the equilateral triangle itself. In other words, an entire<br />
semi-spidron was lurking within the original equilateral triangle, waiting to be<br />
released! Systems of such spidrons are notable for their ability to generate<br />
beautiful tiling patterns in two dimensions (Figure 4.34(b)) and, when folded,<br />
splendidly complex polyhedral shapes in three dimensions (Figure 4.34(c)).<br />
They were invented in 1979 by graphic artist Dániel Erdély as part of a homework<br />
assignment for Ernö Rubik’s (of Rubik’s Cube fame) Theory of Form class<br />
at Budapest University of Art and Design. Possible practical applications of<br />
spidrons include acoustic tiles and shock absorbers for machinery. [242].<br />
1/3
Chapter 5<br />
Mathematical Competitions<br />
As should be abundantly clear from the previous chapters, the equilateral<br />
triangle is a fertile source of mathematical material which requires neither elaborate<br />
mathematical technique nor heavy mathematical machinery. As such,<br />
it has provided grist for the mill of mathematical competitions such as the<br />
American Mathematics Competitions (AMC), USA Mathematical Olympiad<br />
(USAMO) and the International Mathematical Olympiad (IMO).<br />
Problem 1 (AMC 1951). An equilateral triangle is drawn with a side of<br />
length of a. A new equilateral triangle is formed by joining the midpoints of<br />
the sides of the first one, and so on forever. Show that the limit of the sum of<br />
the perimeters of all the triangles thus drawn is 6a. [262, p. 12]<br />
Problem 2 (AMC 1952). Show that the ratio of the perimeter of an equilateral<br />
triangle, having an altitude equal to the radius of a circle, to the perimeter<br />
of an equilateral triangle inscribed in the circle is 2 : 3. [262, p. 20]<br />
Figure 5.1: AMC 1964<br />
129
130 Mathematical Competitions<br />
Problem 3 (AMC 1964). In Figure 5.1, the radius of the circle is equal to<br />
the altitude of the equilateral triangle ABC. The circle is made to roll along<br />
the side AB, remaining tangent to it at a variable point T and intersecting<br />
sides AC and BC in variable points M and N, respectively. Let n be the<br />
number of degrees in arc MTN. Show that, for all permissible positions of the<br />
circle, n remains constant at 60 ◦ . [263, p. 36]<br />
Problem 4 (AMC 1967). The side of an equilateral triangle is s. A circle<br />
is inscribed in the triangle and a square is inscribed in the circle. Show that<br />
the area of the square is s 2 /6. [264, p. 15]<br />
Problem 5 (AMC 1970). An equilateral triangle and a regular hexagon have<br />
equal perimeters. Show that if the area of the triangle is T then the area of the<br />
hexagon is 3T/2. [264, p. 28]<br />
Figure 5.2: AMC 1974<br />
Problem 6 (AMC 1974). In Figure 5.2, ABCD is a unit square and CMN<br />
is an equilateral triangle. Show that the area of CMN is equal to 2 √ 3 − 3<br />
square units. [11, p. 11]<br />
Problem 7 (AMC 1976). Given an equilateral triangle with side of length<br />
s, consider the locus of all points P in the plane of the triangle such that the<br />
sum of the squares of the distances from P to the vertices of the triangle is a<br />
fixed number a. Show that this locus is the empty set if a < s 2 , a single point<br />
if a = s 2 and a circle if a > s 2 . [11, p. 24]
Mathematical Competitions 131<br />
Figure 5.3: AMC 1977<br />
Problem 8 (AMC 1977). In Figure 5.3, each of the three circles is externally<br />
tangent to the other two, and each side of the triangle is tangent to two of the<br />
circles. If each circle has radius ρ then show that the perimeter of the triangle<br />
is ρ · (6 + 6 √ 3). [11, p. 29]<br />
Figure 5.4: AMC 1978<br />
Problem 9 (AMC 1978). In Figure 5.4, if ∆A1A2A3 is equilateral and An+3<br />
is the midpoint of line segment AnAn+1 for all positive integers n, then show<br />
that the measure of ∠A44A45A43 equals 120 ◦ . [11, p. 39]
132 Mathematical Competitions<br />
Figure 5.5: AMC 1981<br />
Problem 10 (AMC 1981). In Figure 5.5, equilateral ∆ABC is inscribed<br />
in a circle. A second circle is tangent internally to the circumcircle at T and<br />
tangent to sides AB and AC at points P and Q, respectively. Show that the<br />
ratio of the length of PQ to the length of BC is 2 : 3. [11, p. 56]<br />
Problem 11 (AMC 1983). Segment AB is a diameter of a unit circle and<br />
a side of an equilateral triangle ABC. The circle also intersects AC and BC<br />
at points D and E, respectively. Show that the length of AE is equal to √ 3.<br />
[22, p. 2]<br />
Figure 5.6: AMC 1988<br />
Problem 12 (AMC 1988). In Figure 5.6, ABC and A ′ B ′ C ′ are equilateral<br />
triangles with parallel sides and the same center. The distance between side<br />
BC and B ′ C ′ is 1 the altitude of ∆ABC. Show that the ratio of the area of<br />
6<br />
∆A ′ B ′ C ′ to the area of ∆ABC is 1 : 4. [22, p. 36]
Mathematical Competitions 133<br />
Figure 5.7: AMC 1988D<br />
Problem 13 (AMC 1988D). In Figure 5.7, a circle passes through vertex<br />
C of equilateral triangle ABC and is tangent to side AB at point F between A<br />
and B. The circle meets AC and BC at D and E, respectively. If AF/FB = p<br />
then show that AD/BE = p 2 . [22, p. 48]<br />
Figure 5.8: AMC 1991<br />
Problem 14 (AMC 1991). In Figure 5.8, equilateral triangle ABC has been<br />
creased and folded so that vertex A now rests at A ′ on BC. If BA ′ = 1 and<br />
A ′ C = 2 then show that PQ = 7√21. [270, p. 22]<br />
20
134 Mathematical Competitions<br />
Figure 5.9: AMC 1992<br />
Problem 15 (AMC 1992). In Figure 5.9, five equilateral triangles, each with<br />
side 2 √ 3, are arranged so they are all on the same side of a line containing<br />
one side of each. Along this line, the midpoint of the base of one triangle is a<br />
vertex of the next. Show that the area of the region of the plane that is covered<br />
by the union of the five triangular regions is equal to 12 √ 3. [270, p. 26]<br />
Figure 5.10: AMC 1995<br />
Problem 16 (AMC 1995). In Figure 5.10, equilateral triangle DEF is inscribed<br />
in equilateral triangle ABC with DE ⊥ BC. Show that the ratio of the<br />
area of ∆DEF to the area of ∆ABC is 1 : 3. [251, p. 5]<br />
Problem 17 (AMC 1998). A regular hexagon and an equilateral triangle<br />
have equal areas. Show that the ratio of the length of a side of the triangle to<br />
the length of a side of the hexagon is √ 6 : 1. [251, p. 27]<br />
Problem 18 (AMC-10 2003). The number of inches in the perimeter of<br />
an equilateral triangle equals the number of square inches in the area of its<br />
circumscribed circle. Show that the radius of the circle is 3 √ 3/π. [99, p. 23]
Mathematical Competitions 135<br />
Figure 5.11: AMC-10 2004<br />
Problem 19 (AMC-10 2004). In Figure 5.11, points E and F are located<br />
on square ABCD so that ∆BEF is equilateral. Show that the ratio of the area<br />
of ∆DEF to that of ∆ABE is 2 : 1. [99, p. 34]<br />
Figure 5.12: AMC-10 2005<br />
Problem 20 (AMC-10 2005). The trefoil shown in Figure 5.12 is constructed<br />
by drawing circular sectors about sides of the congruent equilateral<br />
triangles. Show that if the horizontal base has length 2 then the area of the<br />
. [99, p. 42]<br />
trefoil is 2π<br />
3<br />
Problem 21 (AMC-12 2003a). A square and an equilateral triangle have<br />
the same perimeter. Let A be the area of the circle circumscribed about the<br />
square and B be the area of the circle circumscribed about the triangle. Show<br />
. [323, p. 18]<br />
that A<br />
B<br />
= 27<br />
32
136 Mathematical Competitions<br />
Problem 22 (AMC-12 2003b). A point P is chosen at random in the interior<br />
of an equilateral triangle ABC. Show that the probability that ∆ABP<br />
. [323, p. 20]<br />
has a greater area than each of ∆ACP and ∆BCP is equal to 1<br />
3<br />
Problem 23 (AMC-12 2005). All the vertices of an equilateral triangle lie<br />
on the parabola y = x 2 , and one of its sides has a slope of 2. The x-coordinates<br />
of the three vertices has a sum of m/n, where m and n are relatively prime<br />
positive integers. Show that m + n = 14. [323, p. 46]<br />
Problem 24 (AMC-12 2007a). Point P is inside equilateral ∆ABC. Points<br />
Q, R and S are the feet of the perpendiculars from P to AB, BC and CA,<br />
respectively. Given that PQ = 1, PR = 2 and PS = 3, show that AB = 4 √ 3.<br />
[323, p. 63]<br />
Problem 25 (AMC-12 2007b). Two particles move along the edges of equilateral<br />
∆ABC in the direction A → B → C → A, starting simultaneously and<br />
moving at the same speed. One starts at A and the other starts at the midpoint<br />
of BC. The midpoint of the line segment joining the two particles traces out a<br />
path that encloses a region R. Show that the ratio of the area of R to the area<br />
of ∆ABCis 1 : 16. [323, p. 64]<br />
Figure 5.13: USAMO 1974<br />
Problem 26 (USAMO 1974). Consider the two triangles ∆ABC and ∆PQR<br />
shown in Figure 5.13. In ∆ABC, ∠ADB = ∠BDC = ∠CDA = 120 ◦ . Prove<br />
that x = u + v + w. [196, p. 3]
Mathematical Competitions 137<br />
Figure 5.14: USAMO 2007<br />
Problem 27 (USAMO 2007). Let ABC be an acute triangle with ω, Ω,<br />
and R being its incircle, circumcircle, and circumradius, respectively (Figure<br />
5.14). Circle ωA is tangent internally to Ω at A and tangent externally to ω.<br />
Circle ΩA is tangent internally to Ω at A and tangent internally to ω. Let PA<br />
and QA denote the centers of ωA and ΩA, respectively. Define points PB, QB,<br />
PC, QC analogously. Prove that<br />
8PAQA · PBQB · PCQC ≤ R 3 ,<br />
with equality if an only if triangle ABC is equilateral. [104, p. 28].<br />
Problem 28 (IMO 1961). Let a, b, c be the sides of a triangle, and T its<br />
area. Prove: a 2 + b 2 + c 2 ≥ 4 √ 3T and that equality holds if and only if the<br />
triangle is equilateral. [155, p. 3]<br />
Problem 29 (IMO 1983). Let ABC be an equilateral triangle and E the set<br />
of all points contained in the three segments AB, BC and CA (including A, B<br />
and C). Show that, for every partition of E into two disjoint subsets, at least<br />
one of the two subsets contains the vertices of a right-angled triangle. [195, p.<br />
6]
138 Mathematical Competitions<br />
Problem 30 (IMO Supplemental). Two equilateral triangles are inscribed<br />
in a circle with radius r. Let K be the area of the set consisting of all points<br />
interior to both triangles. Prove that K ≥ r 2√ 3/2. [195, p. 13]<br />
Problem 31 (IMO 1986). Given a triangle A1A2A3 and a point P0 in<br />
the plane, define As = As−3 for all s ≥ 4. Construct a sequence of points<br />
P1, P2, P3, . . . such that Pk+1 is the image of Pk under rotation with center Ak+1<br />
through angle 120 ◦ clockwise (for k = 0, 1, 2, . . .). Prove that if P1986 = P0 then<br />
the triangle A1A2A3 is equilateral. [200, p. 1]<br />
Figure 5.15: IMO 2005<br />
Problem 32 (IMO 2005). Six points are chosen on the sides of an equilateral<br />
triangle ABC: A1 and A2 on BC, B1 and B2 on CA, and C1 and C2 on AB<br />
(Figure 5.15). These points are the vertices of a convex equilateral hexagon<br />
A1A2B1B2C1C2. Prove that lines A1B2, B1C2, and C1A2 are concurrent (at<br />
the center of the triangle). [103, p. 5]<br />
Problem 33 (Austrian-Polish Mathematics Competition 1989). If<br />
each point of the plane is colored either red or blue, prove that some equilateral<br />
triangle has all its vertices the same color. [182, p. 42]
Mathematical Competitions 139<br />
Figure 5.16: All-Union Russian Olympiad 1980<br />
Problem 34 (All-Union Russian Olympiad 1980). A line parallel to<br />
the side AC of equilateral triangle ABC intersects AB at M and BC at P,<br />
thus making ∆BMP equilateral as well (Figure 5.16). Let D be the center of<br />
∆BMP and E be the midpoint of AP. Show that ∆CDE is a 30 ◦ − 60 ◦ − 90 ◦<br />
triangle. [182, p. 125]<br />
Problem 35 (Bulgarian Mathematical Olympiad 1998a). On the sides<br />
of a non-obtuse triangle ABC are constructed externally a square, a regular ngon<br />
and a regular m-gon (m, n > 5) whose centers form an equilateral triangle.<br />
Prove that m = n = 6, and find the angles of triangle ABC. (Answer: The<br />
angles are 90 ◦ , 45 ◦ , 45 ◦ .) [7, p. 9]<br />
Problem 36 (Bulgarian Mathematical Olympiad 1998b). Let ABC be<br />
an equilateral triangle and n > 1 be a positive integer. Denote by S the set<br />
of n − 1 lines which are parallel to AB and divide triangle ABC into n parts<br />
of equal area, and by S ′ the set of n − 1 lines which are parallel to AB and<br />
divide triangle ABC into n parts of equal perimeter. Prove that S and S ′ do<br />
not share a common element. [7, p. 18]<br />
Problem 37 (Irish Mathematical Olympiad 1998). Show that the area<br />
of an equilateral triangle containing in its interior a point P whose distances<br />
from the vertices are 3, 4, and 5 is equal to 9 + 25√ 3<br />
4 . [7, p. 74]<br />
Problem 38 (Korean Mathematical Olympiad 1998). Let D, E, F be<br />
points on the sides BC, CA, AB, respectively, of triangle ABC. Let P, Q, R<br />
be the second intersections of AD, BE, CF, respectively, with the circumcircle<br />
of ABC. Show that AD BE CF + + ≥ 9, with equality if and only if ABC is<br />
PD QE RF<br />
equilateral. [7, p. 84]
140 Mathematical Competitions<br />
Problem 39 (Russian Mathematical Olympiad 1998). A family S of<br />
equilateral triangles in the plane is given, all translates of each other, and any<br />
two having nonempty intersection. Prove that there exist three points such that<br />
every member of S contains one of the points. [7, p. 136]<br />
Problem 40 (Bulgarian Mathematical Olympiad 1999). Each interior<br />
point of an equilateral triangle of side 1 lies in one of six congruent circles of<br />
radius r. Prove that r ≥ √ 3.<br />
[8, p. 33]<br />
10<br />
Problem 41 (French Mathematical Olympiad 1999). For which acuteangled<br />
triangle is the ratio of the shortest side to the inradius maximal? (Answer:<br />
The maximum ratio of 2 √ 3 is attained with an equilateral triangle.) [8,<br />
p. 57]<br />
Problem 42 (Romanian Mathematical Olympiad 1999). Let SABC be<br />
a right pyramid with equilateral base ABC, let O be the center of ABC, and<br />
let M be the midpoint of BC. If AM = 2SO and N is a point on edge SA<br />
such that SA = 25SN, prove that planes ABP and SBC are perpendicular,<br />
where P is the intersection of lines SO and MN. [8, p. 119]<br />
Problem 43 (Romanian IMO Selection Test 1999). Let ABC be an<br />
acute triangle with angle bisectors BL and CM. Prove that ∠A = 60 ◦ if and<br />
only if there exists a point K on BC (K �= B, C) such that triangle KLM is<br />
equilateral. [8, p. 127]<br />
Problem 44 (Russian Mathematical Olympiad 1999). An equilateral<br />
triangle of side length n is drawn with sides along a triangular grid of side<br />
length 1. What is the maximum number of grid segments on or inside the<br />
triangle that can be marked so that no three marked segments form a triangle?<br />
(Answer: n(n + 1).) [8, p. 156]<br />
Problem 45 (Belarusan Mathematical Olympiad 2000). In an equilateral<br />
triangle of n(n+1)<br />
pennies, with n pennies along each side of the triangle,<br />
2<br />
all but one penny shows heads. A “move” consists of choosing two adjacent<br />
pennies with centers A and B and flipping every penny on line AB. Determine<br />
all initial arrangements - the value of n and the position of the coin initially<br />
showing tails - from which one can make all the coins show tails after finitely<br />
many moves. (Answer: For any value of n, the desired initial arrangements<br />
are those in which the coin showing tails is in a corner.) [9, p. 1]<br />
Problem 46 (Romanian Mathematical Olympiad 2000). Let P1P2 · · ·Pn<br />
be a convex polygon in the plane. Assume that, for any pair of vertices Pi, Pj,<br />
there exists a vertex V of the polygon such that ∠PiV Pj = π/3. Show that<br />
n = 3, i.e. show that the polygon is an equilateral triangle. [9, p. 96]
Mathematical Competitions 141<br />
Problem 47 (Turkish Mathematical Olympiad 2000). Show that it is<br />
possible to cut any triangular prism of infinite length with a plane such that<br />
the resulting intersection is an equilateral triangle. [9, p. 147]<br />
Figure 5.17: Hungarian National Olympiad 1987<br />
Problem 48 (Hungarian National Olympiad 1987). Cut the equilateral<br />
triangle AXY from rectangle ABCD in such a way that the vertex X is on<br />
side BC and the vertex Y in on side CD (Figure 5.17). Prove that among the<br />
three remaining right triangles there are two, the sum of whose areas equals<br />
the area of the third. [306, p. 5]<br />
Figure 5.18: Austrian-Polish Mathematics Competition 1993<br />
Problem 49 (Austrian-Polish Mathematics Competition 1993). Let<br />
∆ABC be equilateral. On side AB produced, we choose a point P such that<br />
A lies between P and B. We now denote a as the length of sides of ∆ABC;<br />
r1 as the radius of incircle of ∆PAC; and r2 as the exradius of ∆PBC with<br />
respect to side BC (Figure 5.18). Show that r1 + r2 = a√3. [306, p. 7]<br />
2
142 Mathematical Competitions<br />
Figure 5.19: Iberoamerican Mathematical Olympiad (Mexico) 1993<br />
Problem 50 (Iberoamerican Mathematical Olympiad (Mexico) 1993).<br />
Let ABC be an equilateral triangle and Γ its incircle (Figure 5.19). If D and<br />
E are points of the sides AB and AC, respectively, such that DE is tangent<br />
= 1. [306, p. 9]<br />
to Γ, show that AD<br />
DB<br />
+ AE<br />
EC<br />
Figure 5.20: Mathematical Olympiad of the Republic of China 1994<br />
Problem 51 (Mathematical Olympiad of the Republic of China 1994).<br />
Let ABCD be a quadrilateral with AD = BC and let ∠A+∠B = 120 ◦ . Three<br />
equilateral triangles ∆ACP, ∆DCQ and ∆DBR are drawn on AC, DC and<br />
DB, respectively, away from AB (Figure 5.20). Prove that the three new vertices<br />
P, Q and R are collinear. [306, p. 11]
Mathematical Competitions 143<br />
Figure 5.21: International Mathematical Olympiad (Shortlist) 1996a<br />
Problem 52 (International Mathematical Olympiad (Shortlist) 1996a).<br />
Let ABC be equilateral and let P be a point in its interior. Let the lines AP,<br />
BP, CP meet the sides BC, CA, AB in the points A1, B1, C1, respectively<br />
(Figure 5.21). Prove that<br />
[306, p. 13]<br />
A1B1 · B1C1 · C1A1 ≥ A1B · B1C · C1A.<br />
Figure 5.22: International Mathematical Olympiad (Shortlist) 1996b<br />
Problem 53 (International Mathematical Olympiad (Shortlist) 1996b).<br />
Let ABC be an acute-angled triangle with circumcenter O and circumradius<br />
R. Let AO meet the circle BOC again in A ′ , let BO meet the circle COA<br />
again in B ′ , and let CO meet the circle AOB again in C ′ (Figure 5.22). Prove<br />
that<br />
OA ′ · OB ′ · OC ′ ≥ 8R 3 ,<br />
with equality if and only if ∆ABC is equilateral. [306, p. 13]
144 Mathematical Competitions<br />
Figure 5.23: Nordic Mathematics Competition 1994<br />
Problem 54 (Nordic Mathematics Competition 1994). Let O be a point<br />
in the interior of an equilateral triangle ABC with side length a. The lines<br />
AO, BO and CO intersect the sides of the triangle at the points A1, B1 and<br />
C1, respectively (Figure 5.23). Prove that<br />
[306, p. 15]<br />
|OA1| + |OB1| + |OC1| < a.<br />
Problem 55 (Latvian Mathematical Olympiad 1997). An equilateral<br />
triangle of side 1 is dissected into n triangles. Prove that the sum of squares<br />
of all sides of all triangles is at least 3 and that there is equality if and only if<br />
the triangle can be dissected into n equilateral triangles. [306, p. 29]<br />
Figure 5.24: Irish Mathematical Olympiad 1997<br />
Problem 56 (Irish Mathematical Olympiad 1997). Let ABC be an equilateral<br />
triangle. For a point M inside ABC, let D, E, F be the feet of the<br />
perpendiculars from M onto BC, CA, AB, respectively (Figure 5.24). Show<br />
that the locus of all such points M for which ∠FDE is a right angle is the arc<br />
of the circle interior to ∆ABC subtending 150 ◦ on the line segment BC. [306,<br />
p. 31]
Mathematical Competitions 145<br />
Figure 5.25: Mathematical Olympiad of Moldova 1999a<br />
Problem 57 (Mathematical Olympiad of Moldova 1999a). On the sides<br />
BC and AB of the equilateral triangle ABC, the points D and E, respectively,<br />
are taken such that CD : DB = BE : EA = ( √ 5+1)/2. The straight lines AD<br />
and CE intersect in the point O. The points M and N are interior points of the<br />
segments OD and OC, respectively, such that MN � BC and AN = 2OM.<br />
The parallel to the straight line AC, drawn throught the point O, intersects<br />
the segment MC in the point P (Figure 5.25). Prove that the half-line AP<br />
is the bisectrix of the angle MAN. (Note: This problem is ill-posed in that<br />
AN = 2OM cannot be true if the other conditions are true!) [306, p. 44]<br />
Figure 5.26: Mathematical Olympiad of Moldova 1999b<br />
Problem 58 (Mathematical Olympiad of Moldova 1999b). On the sides<br />
BC, AC and AB of the equilateral triangle ABC, consider the points M, N<br />
and P, respectively, such that AP : PB = BM : MC = CN : NA = λ (Figure<br />
5.26). Show that the circle with diameter AC covers the triangle bounded by<br />
the straight lines AM, BN and CP if and only if 1 ≤ λ ≤ 2. (In the case of<br />
2<br />
concurrent straight lines, the triangle degenerates into a point.) [306, p. 44]
146 Mathematical Competitions<br />
Figure 5.27: Miscellaneous #1<br />
Problem 59 (Miscellaneous #1). Altitude AD of equilateral ∆ABC is a<br />
diameter which intersects AB and AC at E and F, respectively, as in Figure<br />
5.27. Show that the ratio EF : BC = 3 : 4 and that the ratio EB : BD = 1 : 2.<br />
[246, p. 17]<br />
Figure 5.28: Miscellaneous #2<br />
Problem 60 (Miscellaneous #2). Show that the ratio between the area of<br />
a square inscribed in a circle and an equilateral triangle circumscribed about<br />
the same circle is equal to 2√ 3<br />
9 (Figure 5.28). Also, show that the ratio between<br />
the area of a square cirmcumscribed about a circle and an equilateral triangle<br />
inscribed in the same circle is equal to 16√ 3<br />
9 . [246, p. 24]
Mathematical Competitions 147<br />
Figure 5.29: Miscellaneous #3<br />
Problem 61 (Miscellaneous #3). Consider the sequence 1, 1 1 1 , , · · · , , · · ·<br />
2 3 n<br />
and construct the successive-difference triangle shown in Figure 5.29. Prove<br />
that Pascal’s triangle results if we turn the displayed triangle 60◦ clockwise so<br />
that 1 appears at the apex, disregard minus signs, and divide through every row<br />
by its leading entry. [277, pp. 78-79]<br />
Problem 62 (Curiosa #1). If four equilateral triangles be made the sides of<br />
a square pyramid: find the ratio which its volume has to that of a tetrahedron<br />
made of the same triangles. (Answer: Two.) [82, p. 11]<br />
Problem 63 (Curiosa #2). Given two equal squares of side 2, in different<br />
horizontal planes, having their centers in the same vertical line, and so placed<br />
that the sides of each are parallel to the diagonals of the other, and at such a<br />
distance apart that, by joining neighboring vertices, 8 equilateral triangles are<br />
formed: find the volume of the solid thus enclosed. (Answer: 8 4√ 2( √ 2+1)<br />
.) [82,<br />
3<br />
p. 15]
Chapter 6<br />
Biographical Vignettes<br />
In the last decades of the 17th Century, the notable British eccentric, John<br />
Aubrey [248], assembled a collection of short biographies [12] which were edited<br />
and published two centuries later by Andrew Clark (Figure 6.1). Included were<br />
mathematical luminaries such as Briggs, Descartes, Harriot, Oughtred, Pell<br />
and Wallis. In the spirit of Aubrey’s Brief Lives, we conclude our deliberations<br />
on the equilateral triangle with a collection of biographical vignettes devoted<br />
to some of the remarkable characters that we have encountered along our way.<br />
Figure 6.1: Aubrey’s Brief Lives<br />
148
Biographical Vignettes 149<br />
Vignette 1 (Pythagoras of Samos: Circa 569-475 B.C.).<br />
Aristotle attributed the motto “All is number.” to Pythagoras more than<br />
a century after the death of the latter. Since all of Pythagoras’ writings, if<br />
indeed there ever were any, have been lost to us, we have to rely on secondhand<br />
sources written much later for details of his life and teachings [148, 165,<br />
233]. Thus, a grain of skepticism is in order when assessing the accuracy of<br />
these accounts. Pythagoras was born on the island of Samos off the coast of<br />
Ionia (Asia Minor). He had a vast golden birthmark on his thigh which the<br />
Greeks believed to be a sign of divinity. He studied Mathematics with Thales<br />
and Anaximander in the Ionian city of Miletus and traveled widely in his<br />
youth, visiting both Egypt and Babylon and absorbing their knowledge into his<br />
evolving philosophy. He eventually settled in Croton in southern Italy where<br />
he established a commune with his followers. The Pythagorean brotherhood<br />
believed that reality was mathematical in nature and practiced a numerical<br />
mysticism which included the tetraktys discussed in Chapter 1 as well as a<br />
numerical basis for both music and astronomy. Amongst their mathematical<br />
discoveries were irrational numbers, the fact that a polygon with n sides has<br />
sum of interior angles equal to 2n − 4 right angles and sum of exterior angles<br />
equal to four right angles, and the five regular solids (although they knew<br />
how to construct only the tetrahedron, cube and octahedron). They also were<br />
the first to prove the so-called Pythagorean theorem which was known to the<br />
Babylonians 1000 years earlier. Due to political turmoil, the Pythagoreans<br />
were eventually driven from Croton but managed to set up colonies throughout<br />
the rest of Italy and Sicily. Pythagoras died, aged 94, after having returned to<br />
Croton.<br />
Vignette 2 (Plato of Athens: 427-347 B.C.).<br />
Plato was born in Athens and studied under Theodorus and Cratylus who<br />
was a student of Heraclitus [107, 166, 202]. He served in the military during<br />
the Peloponnesian War between Athens and Sparta. After his discharge, he<br />
originally desired a political career but had a change of heart after the execution<br />
of his mentor, Socrates, in 399 B.C. He then traveled widely visiting<br />
Eqypt, Sicily and Italy, where he learned of the teachings of Pythagoras. After<br />
another stint in the military when he was decorated for bravery in battle, he<br />
returned to Athens at age 40 and established his Academy which was devoted<br />
to research and instruction in philosophy and science. Plato believed that<br />
young men so trained would make wiser political leaders. Counted among the<br />
Academy’s graduates were Theaetetus (solid geometry), Eudoxus (doctrine of<br />
proportion and method of exhaustion) and Aristotle (philosophy). Above the<br />
entrance to the Academy stood a sign “Let no one ignorant of Geometry enter<br />
here.”. Plato’s principal writings were his Socratic dialogues wherein he elaborated<br />
upon, among other topics, mathematical ideas such as his Theory of
150 Biographical Vignettes<br />
Forms which gave rise to the mathematical philosophy we now call Platonism.<br />
Through his emphasis on proof, Plato strongly influenced the subsequent development<br />
of Hellenic Mathematics. It was in his Timaeus that he propounded<br />
a mathematical theory of the composition of the universe on the basis of the<br />
five Platonic solids (see Chapter 1). Plato’s Academy flourished until 529 A.D.<br />
when it was closed by Christian Emperor Justinian as a pagan establishment.<br />
At over 900 years, it is the longest known surviving university. Plato died in<br />
Athens, aged 80. Source material for Plato is available in [42, 221].<br />
Vignette 3 (Euclid of Alexandria: Circa 325-265 B.C.).<br />
Little is known of the life of Euclid except that he taught at the Library of<br />
Alexandria in Egypt circa 300 B.C. [165]. When King Ptolemy asked him if<br />
there was an easy way to learn Mathematics, he reportedly replied “There is<br />
no royal road to Geometry!”. It is likely that he studied in Plato’s Academy<br />
in Athens since he was thoroughly familiar with the work of Eudoxus and<br />
Theaetetus which he incorporated into his masterpiece on geometry and number<br />
theory, The Elements [164]. This treatise begins with definitions, postulates<br />
and axioms and then proceeds to thirteen Books. Books one to six<br />
deal with plane geometry (beginning with the construction of the equilateral<br />
triangle, see opening of Chapter 1); Books seven to nine deal with number<br />
theory; Book ten deals with irrational numbers; and Books eleven through<br />
thirteen deal with three-dimensional geometry (ending with the construction<br />
of the regular polyhedra and the proof that there are precisely five of them).<br />
More than one thousand editions of The Elements have been published since<br />
it was first printed in 1482. The opening passages of Book I of the oldest extant<br />
manuscript of The Elements appear in the frontispiece. It was copied by<br />
Stephen the Clerk working in Constantinople in 888 A.D. and it now resides<br />
in the Bodleian Library of Oxford University. Euclid also wrote Conics, a lost<br />
work on conic sections that was later extended by Apollonius of Perga. Source<br />
material for Euclid is available in [164].<br />
Vignette 4 (Archimedes of Syracuse: 287-212 B.C.).<br />
Archimedes is considered by most, if not all, historians of Mathematics to<br />
be one of the greatest (pure or applied) Mathematicians of all time [80, 166]. (I<br />
am told that physicists also feel likewise about him.) He was born in Syracuse,<br />
Sicily, now part of Italy but then an important Greek city-state. As a young<br />
man, he studied with the successors of Euclid in Alexandria but returned to<br />
Syracuse for the remainder of his life. Among his many mathematical accomplishments<br />
were his use of infinitesimals (method of exhaustion) to calculate<br />
areas and volumes, a remarkably accurate approximation to π, and the discovery<br />
and proof that a sphere inscribed in a cylinder has two thirds of the<br />
volume and surface area of the cylinder. He regarded the latter as his greatest
Biographical Vignettes 151<br />
accomplishment and had a corresponding figure commemorating this discovery<br />
placed upon his tomb. The Fields Medal in Mathematics bears both the<br />
portrait of Archimedes and the image of a sphere inscribed in a cylinder. His<br />
achievements in physics include the foundations of hydrostatics and statics,<br />
the explanation of the principle of the lever and the development of the compound<br />
pulley to move large weights. In his own lifetime, he was widely known,<br />
not for his Mathematics, but rather for his mechanical contrivances, especially<br />
his war-machines constructed for his relative, King Hieron of Syracuse. The<br />
Archimedes screw is still in use today for pumping liquids and granulated<br />
solids such as coal and grain. (There is one in downtown Flint, Michigan!)<br />
Many of Archimedes’ greatest treatises are lost to us. For example, it is only<br />
through the writings of Pappus that we know of his investigation of the thirteen<br />
semi-regular polyhedra which now bear his name (see Chapter 1). In<br />
1906, J. L. Heiberg discovered a 10th Century palimpsest containing seven of<br />
his treatises including the previously lost The Method of Mechanical Theorems<br />
wherein this master of antiquity shares with us his secret methods of discovery.<br />
The exciting story of the subsequent disappearance and reappearance of The<br />
Archimedes Codex is told in [226]. He died during the Second Punic War at<br />
the siege of Syracuse, aged 75. He reportedly was killed by a Roman soldier<br />
after Archimedes told him “Do not disturb my circles!”. However, this and<br />
other legends surrounding Archimedes, such as his running naked through the<br />
streets of Syracuse shouting “Eureka!” after a flash of insight while bathing,<br />
must be taken with the proverbial grain of salt. Source material for Archimedes<br />
is available in [168].<br />
Vignette 5 (Apollonius of Perga: Circa 262-190 B.C.).<br />
Apollonius, known as ‘The Great Geometer’, was born in Perga which<br />
was a Greek city of great wealth and beauty located on the southwestern<br />
Mediterranean coast of modern-day Turkey [166, 167]. Very few details of<br />
his life are available but we do know that while still a young man he went<br />
to Alexandria to study under the followers of Euclid and later taught there.<br />
His works were influential in the subsequent development of Mathematics.<br />
For example, his Conics introduced the terminology of parabola, ellipse and<br />
hyperbola. Conics consists of eight books but only the first four, based on a<br />
lost work of Euclid, have survived in Greek while the first seven have survived<br />
in Arabic. The contents of Books five to seven are highly original and believed<br />
to be due primarily to Apollonius himself. Here, he came close to inventing<br />
analytic geometry 1800 years before Descartes. However, he failed to account<br />
for negative magnitudes and, while his equations were determined by curves,<br />
his curves were not determined by equations. Through Pappus, we know of<br />
six lost works by Apollonius. In one of these, Tangencies, he shows how<br />
to construct the circle which is tangent to three given circles (Problem of
152 Biographical Vignettes<br />
Apollonius). Property 16 of Chapter 2 defines Apollonian circles and points.<br />
Ptolemy informs us that he was also one of the founders of Greek mathematical<br />
astronomy, where he used geometrical models to explain planetary motions.<br />
Source material for Apollonius is available in [167].<br />
Vignette 6 (Pappus of Alexandria: Circa 290-350).<br />
Pappus was the last of the great Greek geometers and we know little of<br />
his life except that he was born and taught in Alexandria [166]. The 4th<br />
Century A.D. was a period of general stagnation in mathematical development<br />
(“the Silver Age of Greek Mathematics”). This state of affairs makes Pappus’<br />
accomplishments all the more remarkable. His great work in geometry was<br />
called The Synagoge or The Collection and it is a handbook to be read with<br />
the original works intended to revive the classical Greek geometry. It consists<br />
of eight Books each of which is preceded by a systematic introduction. Book I,<br />
which is lost, was concerned with arithmetic while Book II, which is partially<br />
lost, deals with Apollonius’ method for handling large numbers. Book III<br />
treats problems in plane and solid geometry including how to inscribe each<br />
of the five regular polyhedra in a sphere. Book IV contains properties of<br />
curves such as the spiral of Archimedes and the quadratix of Hippias. Book V<br />
compares the areas of different plane figures all having the same perimeter and<br />
the volumes of different solids all with the same surface area. This Book also<br />
compares the five regular Platonic solids and reveals Archimedes’ lost work on<br />
the thirteen semi-regular polyhedra. Book VI is a synopsis and correction of<br />
some earlier astronomical works. The preface to Book VII contains Pappus’<br />
Problem (a locus problem involving ratios of oblique distances of a point from<br />
a given collection of lines) which later occupied both Descartes and Newton,<br />
as well as Pappus’ Centroid Theorem (a pair of related results concerning the<br />
surface area and volume of surfaces and solids of revolution). Book VII itself<br />
contains Pappus’ Hexagon Theorem (basic to modern projective geometry)<br />
which states that three points formed by intersecting six lines connecting two<br />
sets of three collinear points are also collinear. It also discusses the lost works<br />
of Apollonius previously noted. Book VIII deals primarily with mechanics<br />
but intersperses some questions of pure geometry such as how to draw an<br />
ellipse through five given points. Overall, The Collection is a work of very<br />
great historical importance in the study of Greek geometry. Pappus also wrote<br />
commentaries on the works of Euclid and Ptolemy. Source material for Pappus<br />
is available in [221].<br />
Vignette 7 (Leonardo of Pisa (Fibonacci): 1170-1250).<br />
Leonardo of Pisa, a.k.a. Fibonacci, has been justifiably described as the<br />
most talented Western Mathematician of the Middle Ages [143]. Fibonacci<br />
was born in Pisa, Italy but was educated in North Africa where his father held
Biographical Vignettes 153<br />
a diplomatic post. He traveled widely with his father and became thoroughly<br />
familiar with the Hindu-Arabic numerals and their arithmetic. He returned to<br />
Pisa and published his Liber Abaci (Book of Calculation) in 1202. This most<br />
famous of his works is based upon the arithmetic and algebra that he had<br />
accumulated in his travels and served to introduce the Hindu-Arabic placevalued<br />
decimal system, as well as their numerals, into Europe. An example<br />
from Liber Abaci involving the breeding of rabbits gave rise to the so-called<br />
Fibonacci numbers which he did not discover. (See Property 73 in Chapter 2.)<br />
Simultaneous linear equations are also studied in this work. It also contains<br />
problems involving perfect numbers, the Chinese Remainder Theorem, and the<br />
summation of arithmetic and geometric series. In 1220, he published Practica<br />
geometriae which contains a large collection of geometry problems arranged<br />
into eight chapters together with theorems and proofs based upon the work of<br />
Euclid. This book also includes practical information for surveyors. The final<br />
chapter contains “geometrical subtleties” such as inscribing a rectangle and a<br />
square in an equilateral triangle! In 1225 he published his mathematically most<br />
sophisticated work, Liber quadratorum. This is a book on number theory and<br />
includes a treatment of Pythagorean triples as well as such gems as: “There<br />
are no x, y such that x 2 +y 2 and x 2 −y 2 are both squares” and “x 4 −y 4 cannot<br />
be a square”. This book clearly establishes Fibonacci as the major contributor<br />
to number theory between Diophantus and Fermat. He died in Pisa, aged 80.<br />
Source material for Fibonacci is available in [42, 297].<br />
Vignette 8 (Leonardo da Vinci: 1452-1519).<br />
Leonardo da Vinci was born the illegitimate son of a wealthy Florentine<br />
notary in the Tuscan town of Vinci [44]. He grew up to become one of the<br />
greatest painters of all time and perhaps the most diversely talented person<br />
ever to have lived. A bona fide polymath, he was painter, sculptor, architect,<br />
musician, inventor, anatomist, geologist, cartographer, botanist, writer, engineer,<br />
scientist and Mathematician. In what follows, attention is focused on the<br />
strictly mathematical contributions dispersed amongst his legacy of more than<br />
7,000 surviving manuscript pages. In Chapter 1, I have already described the<br />
circumstances surrounding his illustrations of the Platonic solids for Pacioli’s<br />
De Divina Proportione. The knowledge of the golden section which he gained<br />
through this collaboration is reflected through his paintings. His masterpiece<br />
on perspective, Trattato della Pittura, opens with the injunction “Let no one<br />
who is not a Mathematician read my works.”. This admonition seems more<br />
natural when considered together with the observations of Morris Kline: “It<br />
is no exaggeration to state that the Renaissance artist was the best practicing<br />
Mathematician and that in the fifteenth century he was also the most learned<br />
and accomplished theoretical Mathematician.” [197, p. 127]. Leonardo was<br />
engaged in rusty compass constructions [97, p. 174] and also gave an innovative<br />
congruency-by-subtraction proof of the Pythagorean Theorem [98, p. 29].
154 Biographical Vignettes<br />
He discovered that if a triangle is moved so that one vertex moves along a line<br />
while another vertex moves along a second line then the third vertex describes<br />
an ellipse [322, p. 65] and this observation is the basis of a commercial instrument<br />
for drawing an ellipse using trammels [322, p. 66]. He observed that the<br />
angle between an emerging leaf and its predecessor, known as the divergence, is<br />
a constant and thereby explained the resulting logarithmic spiral arrangement<br />
[139, p. 4]. He was also the first to direct attention to “curves of pursuit” [41,<br />
p. 273]. His architectural studies led him to Leonardo’s Symmetry Theorem<br />
which states that all planar isometries are either rotations or reflections [327,<br />
pp. 66, 99]. The remainder of his mathematical discoveries concerned the areas<br />
of lunes, solids of equal volume, reflection in a sphere, inscription of regular<br />
polygons and centers of gravity [56, pp. 43-60]. His greatest contribution to<br />
geometry came in the latter area where he discovered that the lines joining<br />
the vertices of a tetrahedron with the center of gravity of the opposite faces<br />
all pass through a point, the centroid, which divides each of these medians in<br />
the ratio 3 : 1. These diverse and potent mathematical results have certainly<br />
earned Leonardo the title of Mathematician par excellence! He died at the<br />
castle of Cloux in Amboise, France, aged 67.<br />
Vignette 9 (Niccolò Fontana (Tartaglia): 1499-1557).<br />
Niccolò Fontana was a Mathematician, engineer, surveyor and bookkeeper<br />
who was born in Brescia in the Republic of Venice (now Italy) [144]. Brought<br />
up in dire poverty, he became known as Tartaglia (“The Stammerer”) as a<br />
result of horrific facial injuries which impeded his speech that he suffered in<br />
his youth at the hands of French soldiers. He was widely known during his<br />
lifetime for his participation in many public mathematical contests. He became<br />
a teacher of Mathematics at Verona in 1521 and moved to Venice in 1534 where<br />
he stayed for the rest of his life, except for an 18 month hiatus as Professor<br />
at Brescia beginning in 1548. He is best known for his solution to the cubic<br />
equation sans quadratic term (which first appeared in Cardano’s Ars Magna)<br />
but also is known for Tartaglia’s Formula for the volume of a tetrahedron.<br />
His first book, Nuova scienzia (1551), dealt with the theory and practice of<br />
gunnery. His largest work, Trattato generale di numeri e misure (1556), is<br />
a comprehensive mathematical treatise on arithmetic, geometry, mensuration<br />
and algebra as far as quadratic equations. It is here that he treated the “three<br />
jugs problem” described in Recreation 21 of Chapter 4. He also published the<br />
first Italian translation of Euclid (1543) and the earliest Latin version from the<br />
Greek of some of the principal works of Archimedes (1543). He died at Venice,<br />
aged 58.<br />
Vignette 10 (Johannes Kepler: 1571-1630).<br />
Johannes Kepler was born in the Free Imperial City of Weil der Stadt which<br />
is now part of the Stuttgart Region in the German state of Baden-Württemberg
Biographical Vignettes 155<br />
[105]. As a devout Lutheran, he enrolled at the University of Tübingen in 1589<br />
as a student of theology but also studied mathematics and astronomy under<br />
Michael Mästlin, one of the leading astronomers of the time, who converted<br />
him to the Copernican view of the cosmos. At the end of his university studies<br />
in 1594, he abandoned his plans for ordination (in fact, he was excommunicated<br />
in 1612) and accepted a post teaching Mathematics in Graz. In 1596,<br />
he published Mysterium cosmographicum where he put forth a model of the<br />
solar system based upon inscribing and circumscribing each of the five Platonic<br />
solids by spherical orbs. On this basis, he moved to Prague in 1600 as Tycho<br />
Brahe’s mathematical assistant and began work on compiling the Rudolphine<br />
Tables. In 1601, upon Tycho’s death, he succeeded him as Imperial Mathematician<br />
and the next eleven years proved to be the most productive of his life.<br />
Kepler’s primary obligations were to provide astrological advice to Emperor<br />
Rudolph II and to complete the Rudolphine Tables. In 1604, he published<br />
Astronomiae pars optica where he presented the inverse-square law governing<br />
the intensity of light, treated reflection by flat and curved mirrors and elucidated<br />
the principles of pinhole cameras (camera obscura), as well as considered<br />
the astronomical implications of optical phenomena such as parallax and the<br />
apparent sizes of heavenly bodies. He also considered the optics of the human<br />
eye, including the inverted images formed on the retina. That same year, he<br />
wrote of a “new star” which is today called Kepler’s supernova. In 1609, he<br />
published Astronomia nova where he set out his first two laws of planetary<br />
motion based upon his observations of Mars. In 1611, he published Dioptrice<br />
where he studied the properties of lenses and presented a new telescope design<br />
using two convex lenses, now known as the Keplerian telescope. That same<br />
year, he moved to Linz to avoid religious persecution and, as a New Year’s gift<br />
for his friend and sometimes patron Baron von Wackhenfels, published a short<br />
pamphlet, Strena Seu de Nive Sexangula, where he described the hexagonal<br />
symmetry of snowflakes and posed the Kepler Conjecture about the most efficient<br />
arrangement for packing spheres. Kepler’s Conjecture was solved only<br />
after almost 400 years by Thomas Hales [301]! In 1615, he published a study<br />
of the volumes of solids of revolution to measure the contents of wine barrels,<br />
Nova stereometria doliorum vinariorum, which is viewed today as an ancestor<br />
of the infinitesimal calculus. In 1619, Kepler published his masterpiece, Harmonice<br />
Mundi, which not only contains Kepler’s Third Law, but also includes<br />
the first systematic treatment of tessellations, a proof that there are only thirteen<br />
Archimedean solids (he provided the first known illustration of them as<br />
a set and gave them their modern names), and two new non-convex regular<br />
polyhedra (Kepler’s solids). In 1624 and 1625, he published an explanation<br />
of how logarithms worked and he included eight-figure logarithmic tables with<br />
the Rudolphine Tables which were finally published in 1628. In that year, he<br />
left the service of the Emperor and became an advisor to General Wallenstein.
156 Biographical Vignettes<br />
He fell ill while visiting Regensburg, Bavaria and died, aged 58. (His tomb was<br />
destroyed in the course of the Thirty Years’ War.) Somnium (1634), which<br />
was published posthumously, aimed to show the feasibility of a non-geocentric<br />
system by describing what practicing astronomy would be like from the perspective<br />
of another planet. Kepler’s work on regular and semiregular tilings<br />
of the plane was mentioned in Chapter 1 as was the likelihood that he was a<br />
Rosicrucian. Source material for Kepler is available in [42].<br />
Vignette 11 (René Descartes: 1596-1650).<br />
René Descartes, the Father of Modern Philosophy, was born in La Haye<br />
en Touraine (now renamed Descartes), Indre-et-Loire, France [3]. He was educated<br />
at the Jesuit College of La Flèche in Anjou until 1612. He received a<br />
law degree from the University of Poitiers in 1616 and then enlisted in the military<br />
school at Breda in the Dutch Republic. Here he met the Dutch scientist<br />
Isaac Beeckman with whom he began studying mechanics and Mathematics in<br />
1618, then, in 1619, he joined the Bavarian army. From 1620 to 1628, he wandered<br />
throughout Europe, spending time in Bohemia (1620), Hungary (1621),<br />
Germany, Holland and France (1622-23). In 1623, he met Marin Mersenne in<br />
Paris, an important contact which kept him in touch with the scientific world<br />
for many years. From Paris, he travelled to Italy where he spent some time in<br />
Venice, then he returned to France again (1625). In 1628, he chose to settle<br />
down in Holland for the next twenty years. In 1637, he published a scientific<br />
treatise, Discours de la méthode, which included a treatment of the tangent<br />
line problem which was to provide the basis for the calculus of Newton and<br />
Leibniz. It also contained among its appendices his masterpiece on analytic<br />
geometry, La Géométrie, which includes Descartes’ Rule of Signs for determining<br />
the number of positive and negative real roots of a polynomial. In another<br />
appendix, on optics, he independently discovered Snell’s law of reflection. In<br />
1644, he published Principia Philosophiae where he presented a mathematical<br />
foundation for mechanics that included a vortex theory as an alternative to<br />
action at a distance. In 1649, Queen Christina of Sweden persuaded him to<br />
move to Stockholm, where he died of pneumonia, aged 53. Descartes’ proficiency<br />
at bare-knuckled brawling is revealed by his response to criticism of his<br />
work by Fermat: he asserted euphemistically that he was “full of shit” [169, p.<br />
38]. Descartes’ role in the discovery of the polyhedral formula was mentioned<br />
in Chapter 1 as was the likelihood that he was a Rosicrucian. Source material<br />
for Descartes is available in [42, 77, 221, 287, 297].<br />
Vignette 12 (Pierre de Fermat: 1601-1665).<br />
Pierre de Fermat, lawyer and Mathematician, was born in Beaumont-de-<br />
Lomagne, France [216]. He began his studies at the University of Toulouse
Biographical Vignettes 157<br />
before moving to Bordeaux where he began his first serious mathematical researches.<br />
He received his law degree from the University of Orleans in 1631<br />
and received the title of councillor at the High Court of Judicature in Toulouse,<br />
which he held for the rest of his life. For the remainder of his life, he lived<br />
in Toulouse while also working in his home town of Beaumont-de-Lomagne<br />
and the nearby town of Castres. Outside of his official duties, Fermat was<br />
preoccupied with Mathematics and he communicated his results in letters to<br />
his friends, especially Marin Mersenne, often with little or no proof of his theorems.<br />
He developed a method for determining maxima, minima and tangents<br />
to various curves that was equivalent to differentiation. He also developed a<br />
technique for finding centers of gravity of various plane and solid figures that<br />
led him to further work in quadrature. In number theory, he studied Pell’s<br />
equation, perfect numbers (where he discovered Fermat’s Little Theorem),<br />
amicable numbers and what would later become known as Fermat numbers.<br />
He invented Fermat’s Factorization Method and the technique of infinite descent<br />
which he used to prove Fermat’s Last Theorem for n = 4. (Fermat’s<br />
Last Theorem [19] was finally resolved, after more that 350 years, by Andrew<br />
Wiles [283]!) He drew inspiration from Diophantus, but was interested only<br />
in integer solutions to Diophantine equations, and he looked for all possible<br />
solutions. Through his correspondence with Pascal, he helped lay the fundamental<br />
groundwork for the theory of probability. Fermat’s Principle of Least<br />
Time (which he used to derive Snell’s Law) was the first variational principle<br />
enunciated in more than 1,500 years. See Property 12 for a definition of the<br />
Fermat point and Property 39 for a discussion of Fermat’s Polygonal Number<br />
Conjecture. He died in Castres, France, aged 57. Source material for Fermat<br />
is available in [42, 287, 297].<br />
Vignette 13 (Evangelista Torricelli: 1608-1647).<br />
Evangelista Torricelli was a physicist and Mathematician born in Faenza,<br />
part of the Papal States [144]. He went to Rome in 1627 to study science<br />
under the Benedictine Benedetto Castelli, Professor of Mathematics at the<br />
Collegio della Sapienza. He then served for nine years as secretary to Giovanni<br />
Ciampoli, a friend of Galileo. In 1641, Torricelli moved to Arcetri where<br />
Galileo was under house arrest and worked with him and Viviani for a few<br />
months prior to Galileo’s death. He was then appointed to succeed Galileo<br />
as Court Mathematician to Grand Duke Ferdinando II de’ Medici of Tuscany.<br />
He held this post until his death living in the ducal palace in Florence. In<br />
1644, the Opera geometrica appeared, his only work to be published during<br />
his lifetime. He examined the three dimensional figures obtained by rotating<br />
a regular polygon about an axis of symmetry, computed the center of gravity<br />
of the cycloid and extended Cavalieri’s (another pupil of Castelli) method of<br />
indivisibles. Torricelli was the first person to create a sustained vacuum or to
158 Biographical Vignettes<br />
give the correct scientific explanation of the cause of the wind (differences of<br />
air temperature and density) and he discovered the principle of the mercury<br />
barometer. He was a skilled lens grinder and made excellent telescopes and<br />
microscopes. The Fermat-Torricelli Problem/Point has already been discussed<br />
at length in Property 12 of Chapter 2. (This problem was solved by Torricelli<br />
and Cavalieri for triangles of less than 120 ◦ and the general case was solved by<br />
Viviani.) Torricelli’s Trumpet (Gabriel’s Horn) is a figure with infinite surface<br />
area yet finite volume. Torricelli’s Law/Theorem relates the speed of fluid<br />
flowing out of an opening to the height of the fluid above the opening (v =<br />
√ 2gh). Torricelli’s Equation provides the final velocity of an object moving<br />
with constant acceleration without having a known time interval (v 2 f = v2 i +<br />
2a∆d). He died in Florence, aged 39, shortly after having contracted typhoid<br />
fever. Viviani agreed to prepare his unpublished materials for posthumous<br />
publication but he failed to accomplish this task, which was not completed until<br />
1944 nearly 300 years after Torricelli’s death. Source material for Torricelli is<br />
available in [297].<br />
Vignette 14 (Vincenzo Viviani: 1622-1703).<br />
Vincenzo Viviani, the last pupil of Galileo, was born in Florence, Italy<br />
[144]. His exceptional mathematical abilities brought him to the attention<br />
of Grand Duke Ferdinando II de’ Medici of Tuscany in 1638 who introduced<br />
him to Galileo. In 1639, at the age of 17, he became Galileo’s assistant at<br />
Arcetri until the latter’s death in 1642. During this period, he met Torricelli<br />
and they later became collaborators on the development of the barometer.<br />
(He was Torricelli’s junior colleague not his student although he collected and<br />
arranged his works after the latter’s death.) The Grand Duke then appointed<br />
him Mathematics teacher at the ducal court and engaged him as an engineer<br />
with the Uffiziali dei Fiumi, a position he held for the rest of his life. From<br />
1655 to 1656, he edited the first edition of Galileo’s collected works and he also<br />
wrote the essay Life of Galileo which was not published in his lifetime. In 1660,<br />
he and Giovanni Alfonso Borelli conducted an experiment involving timing the<br />
difference between seeing the flash and hearing the sound of a cannon shot at<br />
a distance which provided an accurate determination of the speed of sound. In<br />
1661, he experimented with rotation of pendula, 190 years before the famous<br />
demonstration by Foucault. In 1666, the Grand Duke appointed him Court<br />
Mathematician. Throughout his life, one of his main interests was ancient<br />
Greek Mathematics and he published reconstructions of lost works of Euclid<br />
and Apollonius and also translated a work of Archimedes into Italian. He<br />
calculated the tangent to the cycloid and also contributed to constructions<br />
involving angle trisection and duplication of the cube. Viviani’s Theorem has<br />
been previously described in Property 8 of Chapter 2. Viviani’s Curve is a<br />
space curve obtained by intersecting a sphere with a circular cylinder tangent
Biographical Vignettes 159<br />
to the sphere and to one of its diameters, a sort of spherical figure eight [38].<br />
Viviani proposed the architectural problem (“Florentine enigma”): Build on<br />
a hemispherical cupola four equal windows of such a size that the remaining<br />
surface can be exactly squared. The Viviani Window is a solution: The four<br />
windows are the intersections of a hemisphere of radius a with two circular<br />
cylinders of radius a<br />
2<br />
that have in common only a ruling containing a diameter<br />
of the hemisphere [38]. He provided the complete solution to the Fermat-<br />
Torricelli Problem (see Property 12 in Chapter 2). In 1687, he published a<br />
book on engineering, Discorso and, upon his death in Florence, aged 81, he left<br />
an almost completed work on the resistance of solids which was subsequently<br />
completed and published by Luigi Guido Grandi.<br />
Vignette 15 (Blaise Pascal: 1623-1662).<br />
Blaise Pascal, philosopher, physicist and Mathematician, was born in Clermont,<br />
Auvergne, France [25, 40]. At age 14, he began to accompany his father<br />
to Mersenne’s meetings of intellectuals which included Roberval and Desargues.<br />
Except for this influence, he was essentially self-taught. At one such<br />
meeting, at the age of sixteen, he presented a number of theorems in projective<br />
geometry, including Pascal’s Mystic Hexagram Theorem. In 1639, he and<br />
his family moved to Rouen and, in 1640, Pascal published his first work, Essay<br />
on Conic Sections. Pascal invented the first digital calculator, the Pascaline,<br />
to help his father with his work collecting taxes. In 1647, he published New<br />
Experiments Concerning Vacuums where he argued for the existence of a pure<br />
vacuum and observed that atmospheric pressure decreases with height thereby<br />
deducing that a vacuum existed above the atmosphere. In 1653, he published<br />
Treatise on the Equilibrium of Liquids in which he introduced what is now<br />
known as Pascal’s Law of Pressure. (He also invented the hydraulic press and<br />
the syringe.) In a lost work (we know of its contents because Leibniz and<br />
Tschirnhaus had made notes from it), The Generation of Conic Sections, he<br />
presented important theorems in projective geometry. His 1653 work, Treatise<br />
on the Arithmetical Triangle, was to lead Newton to his discovery of the generalized<br />
binomial theorem for fractional and negative powers. (Pascal’s triangle<br />
is described in more detail in Property 61 of Chapter 2.) In the summer of<br />
1654, he exchanged letters with Fermat where they laid the foundations of<br />
the theory of probability. In the fall of the same year, he had a near-death<br />
experience which led him to devote his remaining years to religious pursuits,<br />
specifically Jansenism. (At which time, he devised Pascal’s Wager.) Thus,<br />
with the exception of a 1658 study of the quadrature of the cycloid, his scientific<br />
and mathematical investigations were concluded, at age 31. The SI unit of<br />
pressure and a programming language are named after him. He died in intense<br />
pain, aged 39, in Paris, France, after a malignant growth in his stomach spread<br />
to his brain. Source material for Pascal is available in [42, 287, 297].
160 Biographical Vignettes<br />
Vignette 16 (Isaac Newton: 1643-1727).<br />
Isaac Newton was born in the manor house of Woolsthorpe, near Grantham<br />
in Lincolnshire, England [298, 324, 328]. Because England had not yet adopted<br />
the Gregorian calender, his birth was recorded as Christmas Day, 25 December<br />
1642. While he showed some mechanical ability as a young man, his mathematical<br />
precocity did not appear until he was a student at Trinity College,<br />
Cambridge where he enrolled in 1661. He was elected a scholar in 1664 and<br />
received his bachelor’s degree in 1665. When the University closed in the summer<br />
of 1665 due to the plague, Newton returned to Woolsthorpe for the next<br />
two years, during which time he developed calculus, performed optical experiments<br />
and discovered the universal law of gravitation. When the University<br />
opened again in 1667, he was elected to a minor fellowship at Trinity College<br />
but, after being awarded his master’s degree, he was elected to a major fellowship<br />
in 1668. In 1669, Isaac Barrow, the Lucasian Professor of Mathematics,<br />
began to circulate Newton’s tract on infinite series, De Analysi, so that, when<br />
he stepped down that same year, Newton was chosen to fill the Lucasian Chair.<br />
Newton then turned his attention to optics and constructed a reflecting telescope<br />
which led to his election as a Fellow of the Royal Society. In 1672,<br />
he published his first scientific paper on light and colour where he proposed<br />
a corpuscular theory. This received heavy criticism by Hooke and Huyghens<br />
who favored a wave theory. Because of the ensuing controversy, Newton became<br />
very reluctant to publish his later discoveries. However, in 1687, Halley<br />
convinced him to publish his greatest work Philosophiae naturalis principia<br />
mathematica. Principia contained his three laws of motion and the inverse<br />
square law of gravitation, as well as their application to orbiting bodies, projectiles,<br />
pendula, free-fall near the Earth, the eccentric orbits of comets, the<br />
tides and their variations, the precession of the Earth’s axis, and the motion<br />
of the Moon as perturbed by the gravity of the Sun. It has rightfully been<br />
called the greatest scientific treatise ever written! His other mathematical<br />
contributions include the generalized binomial theorem, Newton’s method for<br />
approximating the roots of a function, Newton’s identities relating the roots<br />
and coefficients of polynomials, the classification of cubic curves, the theory<br />
of finite differences and the Newton form of the interpolating polynomial. His<br />
other contributions to physics include the formulation of his law of cooling and<br />
a study of the speed of sound. However, to maintain a proper perspective, one<br />
must bear in mind that Newton wrote much more on Biblical and alchemical<br />
topics than he ever did in physics and Mathematics! In 1693, he left Cambridge<br />
to become first Warden and then Master of the Mint. In 1703 he was elected<br />
President of the Royal Society and was re-elected each year until his death. In<br />
1705, he was knighted by Queen Anne (the first scientist to be so honored). In<br />
his later years, he was embroiled in a bitter feud with Leibniz over priority for<br />
the invention of calculus [169]. He died in his sleep in London, aged 84, and
Biographical Vignettes 161<br />
was buried in Westminster Abbey. Chapter 1 noted how Newton patterned his<br />
Principia after Euclid’s The Elements and alluded to his alchemical activities.<br />
Source material for Newton is available in [42, 221, 227, 287, 297].<br />
Vignette 17 (Leonhard Euler: 1707-1783).<br />
Leonhard Euler was born in Basel, Switzerland and entered the University<br />
of Basel at age 14 where he received private instruction in Mathematics from<br />
the eminent Mathematician Johann Bernoulli [102]. He received his Master<br />
of Philosophy in 1723 with a dissertation comparing the philosophies of<br />
Descartes and Newton. In 1726, he completed his Ph.D. with a dissertation on<br />
the propagation of sound and published his first paper on isochronous curves<br />
in a resisting medium. In 1727, he published another paper on reciprocal<br />
trajectories [265, pp. 6-7] and he also won second place in the Grand Prize<br />
competition of the Paris Academy for his essay on the optimal placement of<br />
masts on a ship. He subsequently won first prize twelve times in his career.<br />
Also in 1727, he wrote a classic paper in acoustics and accepted a position in<br />
the mathematical-physical section of the St. Petersburg Academy of Sciences<br />
where he was a colleague of Daniel Bernoulli, son of Euler’s teacher in Basel.<br />
During this first period in St. Petersburg, his research was focused on number<br />
theory, differential equations, calculus of variations and rational mechanics. In<br />
1736-37, he published his book Mechanica which presented Newtonian dynamics<br />
in the form of mathematical analysis and, in 1739, he laid a mathematical<br />
foundation for music. In 1741, he moved to the Berlin Academy. During the<br />
twenty five years that he spent in Berlin, he wrote 380 articles and published<br />
books on calculus of variations, the calculation of planetary orbits, artillery<br />
and ballistics, analysis, shipbuilding and navigation, the motion of the moon,<br />
lectures on differential calculus and a popular scientific publication, Letters<br />
to a Princess of Germany. In 1766, he returned to St. Petersburg where he<br />
spent the rest of his life. Although he lost his sight, more than half of his total<br />
works date to this period, primarily in optics, algebra and lunar motion, and<br />
also an important work on insurance. To Euler, we owe the notation f(x), e,<br />
i, π, Σ for sums and ∆ n y for finite differences. He solved the Basel Problem<br />
Σ(1/n 2 ) = π 2 /6, proved the connection between the zeta function and the sequence<br />
of prime numbers, proved Fermat’s Last Theorem for n = 3, and gave<br />
the formula e ıθ = cosθ +ı·sin θ together with its special case e ıπ +1 = 0. The<br />
list of his important discoveries that I have not included is even longer! As<br />
Laplace advised, “Read Euler, read Euler, he is the master of us all.”. Euler<br />
was the most prolific Mathematician of all time with his collected works filling<br />
between 60-80 quarto volumes. See Chapter 1 for a description of Euler’s role<br />
in the discovery of the Polyhedral Formula, Property 28 for the definition of<br />
the Euler line, Property 47 for the statement of Euler’s inequality and Recreation<br />
25 for his investigation of the Knight’s Tour. He has been featured on the
162 Biographical Vignettes<br />
Swiss 10-franc banknote and on numerous Swiss, German and Russian stamps.<br />
As part of their Tercentenary Euler Celebration, the Mathematical Association<br />
of America published a five volume tribute [265, 89, 266, 27, 32]. He died<br />
in St. Petersburg, Russia, aged 76. Source material for Euler is available in<br />
[42, 287, 297].<br />
Vignette 18 (Gian Francesco Malfatti: 1731-1807).<br />
Gian Francesco Malfatti was born in Ala, Trento, Italy but studied at<br />
the College of San Francesco Saverio in Bologna under Francesco Maria Zanotti,<br />
Gabriele Manfredi and Vincenzo Riccati (Father of Hyperbolic Functions)<br />
[144]. He then went to Ferrara in 1754, where he founded a school of Mathematics<br />
and physics. In 1771, when the University of Ferrara was reestablished,<br />
he was appointed Professor of Mathematics, a position he held for approximately<br />
thirty years. In 1770, he worked on the solution of the quintic equation<br />
where he introduced the Malfatti resolvent. In 1781, he demonstrated that<br />
the lemniscate has the property that a point mass moving on it under gravity<br />
goes along any arc of the curve in the same time as it traverses the subtending<br />
chord. In 1802 he gave the first, brilliant solution of the problem which bears<br />
his name: Describe in a triangle three circles that are mutually tangent, each<br />
of which touches two sides of the triangle (see Property 70 in Chapter 2). He<br />
also made fundamental contributions to probability, mechanics, combinatorial<br />
analysis and to the theory of finite difference equations. He died in Ferrara,<br />
aged 76.<br />
Vignette 19 (Joseph-Louis Lagrange: 1736-1813).<br />
Joseph-Louis Lagrange was born in Turin, Italy and educated at the College<br />
of Turin where he showed little interest in Mathematics until reading,<br />
at age seventeen, a paper by Edmond Halley on the use of algebra in optics<br />
[151]. He then devoted himself to the study of Mathematics. Although he<br />
was essentially self-taught and did not have the benefit of studying with leading<br />
Mathematicians, he was made a Professor of Mathematics at the Royal<br />
Artillery School in Turin at age nineteen. His first major work was on the<br />
tautochrone (the curve on which a particle will always arrive at a fixed point<br />
in the same time independent of its initial position) where he discovered a<br />
method for extremizing functionals which became one of the cornerstones of<br />
calculus of variations. In 1757, Lagrange was one of the founding members<br />
of what was to become the Royal Academy of Sciences in Turin. Over the<br />
next few years, he published diverse papers in its transactions on calculus of<br />
variations (including Lagrange multipliers), calculus of probabilities and foundations<br />
of mechanics based upon the Principle of Least Action. He made a<br />
major study of the propagation of sound where he investigated the vibrating<br />
string using a discrete mass model with the number of masses approaching
Biographical Vignettes 163<br />
infinity. He also studied the integration of differential equations and fluid mechanics<br />
where he introduced the Lagrangian function. In 1764, he submitted<br />
a prize essay to the French Academy of Sciences on the libration of the moon<br />
containing an explanation as to why the same face is always turned towards<br />
Earth which utilized the Principle of Virtual Work and the idea of generalized<br />
equations of motion. In 1766, Lagrange succeeded Euler as the Director of<br />
Mathematics at the Berlin Academy where he stayed until 1787. During the<br />
intervening 20 years, he published a steady stream of top quality papers and<br />
regularly won the prize from the Académie des Sciences in Paris. These papers<br />
covered astronomy, stability of the solar system, mechanics, dynamics, fluid<br />
mechanics, probability and the foundation of calculus. His work on number<br />
theory in Berlin included the Four Squares Theorem and Wilson’s Theorem<br />
(n is prime if and only if (n − 1)! + 1 is divisible by n). He also made a<br />
fundamental investigation of why equations of degree up to 4 can be solved<br />
by radicals and studied permutations of their roots which was a first step in<br />
the development of group theory. However, his greatest achievement in Berlin<br />
was the preparation of his monumental work Traité de mécanique analytique<br />
(1788) which presented from a unified perspective the various principles of mechanics,<br />
demonstrating their connections and mutual dependence. This work<br />
transformed mechanics into a branch of mathematical analysis. In 1787, he<br />
left Berlin to accept a non-teaching post at the Académie des Sciences in Paris<br />
where he stayed for the rest of his career. He was a member of the committee to<br />
standardize weights and measures that recommended the adoption of the metric<br />
system and served on the Bureau des Longitudes which was charged with<br />
the improvement of navigation, the standardization of time-keeping, geodesy<br />
and astronomical observation. His move to Paris signalled a marked decline<br />
in his mathematical productivity with his single notable achievement being<br />
his work on polynomial interpolation. See Property 39 for a description of<br />
his contribution to the Polygonal Number Theorem and Applications 7&8 for<br />
a summary of his research on the Three-Body Problem. He died and was<br />
buried in the Panthéon in Paris, aged 77, before he could finish a thorough<br />
revision of Mécanique analytique. Source material for Lagrange is available in<br />
[23, 42, 297].<br />
Vignette 20 (Johann Karl Friedrich Gauss: 1777-1855).<br />
Karl Friedrich Gauss, Princeps mathematicorum, was born to poor workingclass<br />
parents in Braunschweig in the Electorate of Brunswick-Lüneburg of the<br />
Holy Roman Empire now part of Lower Saxony, Germany [36, 90, 160]. He<br />
was a child prodigy, correcting his father’s financial calculations at age 3 and<br />
discovering the sum of an arithmetic series in primary school. His intellectual<br />
abilities attracted the attention and financial support of the Duke of Braunschweig,<br />
who sent him to the Collegium Carolinum (now Technische Univer-
164 Biographical Vignettes<br />
sität Braunschweig), which he attended from 1792 to 1795, and to the University<br />
of Göttingen from 1795 to 1798. His first great breakthrough came in 1796<br />
when he showed that any regular polygon with a number of sides which is a<br />
Fermat prime can be constructed by compass and straightedge. This achievement<br />
led Gauss to choose Mathematics, which he termed the Queen of the<br />
Sciences, as his life’s vocation. Gauss was so enthused by this discovery that<br />
he requested that a regular heptadecagon (17 sided polygon) be inscribed on his<br />
tombstone, a request that was not fulfilled because of the technical challenge<br />
which it posed. He returned to Braunschweig in 1798 without a degree but received<br />
his doctorate in abstentia from the University of Helmstedt in 1799 with<br />
a dissertation on the Fundamental Theorem of Algebra under the nominal supervision<br />
of J. F. Pfaff. With the Duke’s stipend to support him, he published,<br />
in 1801, his magnum opus Disquisitiones Arithmeticae, a work which he had<br />
completed in 1798 at age 21. Disquisitiones Arithmeticae summarized previous<br />
work in a systematic way, resolved some of the most difficult outstanding<br />
questions, and formulated concepts and questions that set the pattern of research<br />
for a century and that still have significance today. It is here that he<br />
introduced modular arithmetic and proved the law of quadratic reciprocity as<br />
well as appended his work on constructions with compass and straightedge.<br />
In this same year, 1801, he predicted the orbit of Ceres with great accuracy<br />
using the method of least squares, a feat which brought him wide recognition.<br />
With the Duke’s death, he left Braunschweig in 1807 to take up the position of<br />
director of the Göttingen observatory, a post which he held for the rest of his<br />
life. In 1809, he published Theoria motus corporum coelestium, his two volume<br />
treatise on the motion of celestial bodies. His involvement with the geodetic<br />
survey of the state of Hanover (when he invented the heliotrope) led to his<br />
interest in differential geometry. Here, he contributed the notion of Gaussian<br />
curvature and the Theorema Egregium which informally states that the curvature<br />
of a surface can be determined entirely by measuring angles and distances<br />
on the surface, i.e. curvature of a two-dimensional surface does not depend<br />
on how the surface is embedded in three-dimensional space. With Wilhelm<br />
Weber, he investigated terrestrial magnetism, discovered the laws of electric<br />
circuits, developed potential theory and invented the electromechanical telegraph.<br />
His work for the Göttingen University widows’ fund is considered part<br />
of the foundation of actuarial science. Property 38 describes the equilateral<br />
triangle in the Gauss plane and Property 39 states Gauss’ Theorem on Triangular<br />
Numbers. Although not enamored with teaching, he counted amongst<br />
his students such luminaries as Bessel, Dedekind and Riemann. The CGS unit<br />
of magnetic induction is named for him and his image was featured on the<br />
German Deutschmark as well as on three stamps. Gauss was not a prolific<br />
writer, which is reflected in his motto Pauca sed matura (“Few, but ripe”).<br />
What he did publish may best be described as terse, in keeping with his belief
Biographical Vignettes 165<br />
that a skilled artisan should always remove the scaffolding after a masterpiece<br />
is finished. His personal diaries contain several important mathematical discoveries,<br />
such as non-Euclidean geometry, that he had made years or decades<br />
before his contemporaries published them. He died, aged 77, in Göttingen in<br />
the Kingdom of Hanover. His brain was preserved and was studied by Rudolf<br />
Wagner who found highly developed convolutions present, perhaps accounting<br />
for his titanic intellect. His body is interred in the Albanifriedhof cemetery<br />
[304, p. 59] and, in 1995, the present author made a pilgrimage there and<br />
was only too glad to remove the soda pop cans littering this holy shrine of<br />
Mathematics! Source material for Gauss is available in [23, 42, 221, 287, 297].<br />
Vignette 21 (Jakob Steiner: 1796-1863).<br />
Jacob Steiner, considered by many to have been the greatest pure geometer<br />
since Apollonius of Perga, was born in the village of Utzenstorf just north of<br />
Bern, Switzerland [144]. At age 18, he left home to attend J. H. Pestalozzi’s<br />
school at Yverdon where the educational methods were child-centered and<br />
based upon individual learner differences, sense perception and the student’s<br />
self-activity. In 1818, he went to Heidelberg where he attended lectures on combinatorial<br />
analysis, differential and integral calculus and algebra, and earned<br />
his living giving private Mathematics lessons. In 1821, he traveled to Berlin<br />
where he first supported himself through private tutoring before obtaining a<br />
license to teach Mathematics at a Gymnasium. In 1834, he was appointed<br />
Extraordinary Professor of Mathematics at the University of Berlin, a post<br />
he held until his death. In Berlin, he made the acquaintance of Niels Abel,<br />
Carl Jacobi and August Crelle. Steiner became an early contributor to Crelle’s<br />
Jounal, which was the first journal entirely devoted to Mathematics. In 1826,<br />
the premier issue contained a long paper by Steiner (the first of 62 which were<br />
to appear in Crelle’s Journal) that introduced the power of a point with respect<br />
to a circle, the points of similitude of circles and his principle of inversion.<br />
This paper also considers the problem: What is the maximum number of parts<br />
into which a space can be divided by n planes? (Answer: n3 +5n+6.)<br />
In 1832,<br />
6<br />
Steiner published his first book, Systematische Entwicklung der Abhangigkeit<br />
geometrischer Gestalten voneinander, where he gives explicit expression to his<br />
approach to Mathematics: “The present work is an attempt to discover the<br />
organism through which the most varied spatial phenomena are linked with<br />
one another. There exist a limited number of very simple fundamental relationships<br />
that together constitute the schema by means of which the remaining<br />
theorems can be developed logically and without difficulty. Through the proper<br />
adoption of the few basic relations one becomes master of the entire field.”. He<br />
was one of the greatest contributors to projective geometry (Steiner surface<br />
and Steiner Theorem). Then, there is the beautiful Poncelet-Steiner Theorem<br />
which shows that only one given circle and a straightedge are required
166 Biographical Vignettes<br />
for Euclidean constructions. He also considered the problem: Of all ellipses<br />
that can be circumscribed about (inscribed in) a given triangle, which one<br />
has the smallest (largest) area? (Today, these ellipses are called the Steiner<br />
ellipses.) Steiner disliked algebra and analysis and advocated an exclusively<br />
synthetic approach to geometry. See Property 70 of Chapter 2 for his role in<br />
solving Malfatti’s Problem and Application 20 of Chapter 3 for the application<br />
of Steiner triple-systems to error-correcting codes. He died in Bern, aged 67.<br />
Source material for Steiner is available in [287].<br />
Vignette 22 (Joseph Bertrand: 1822-1900).<br />
Joseph Bertrand was a child prodigy who was born in Paris, France and<br />
whose early career was guided by his uncle, the famed physicist and Mathematician<br />
Duhamel [144]. (He also had familial connections to Hermite, Picard<br />
and Appell.) He began attending lectures at l’ École Polytechnique at age<br />
eleven and was awarded his doctorate at age 17 for a thesis in thermodynamics.<br />
At this same time, he published his first paper on the mathematical theory<br />
of electricity. In 1842, he was badly injured in a train crash and suffered a<br />
crushed nose and facial scars which he retained throughout his life. Early in<br />
his career, he published widely in mathematical physics, mathematical analysis<br />
and differential geometry. He taught at a number of institutions in France<br />
until becoming Professor of Analysis at Collège de France in 1862. In 1845,<br />
he conjectured that there is at least one prime between n and 2n − 2 for every<br />
n > 3. This conjecture was proved by Chebyshev in 1850. In 1845, he made<br />
a major contribution to group theory involving subgroups of low index in the<br />
symmetric group. He was famed as the author of textbooks on arithmetic,<br />
algebra, calculus, thermodynamics and electricity. His book Calcul des probabilitiés<br />
(1888) contains Bertrand’s Paradox which was described in Recreation<br />
11 of Chapter 4. This treatise greatly influenced Poincaré’s work on this same<br />
topic. Bertrand was elected a member of the Paris Academy of Sciences in<br />
1856 and served as its Permanent Secretary from 1874 to the end of his life.<br />
He died in Paris, aged 78.<br />
Vignette 23 (Georg Friedrich Bernhard Riemann: 1826-1866).<br />
Bernhard Riemann was born in Breselenz, a village near Dannenberg in<br />
the Kingdom of Hanover in what is today the Federal Republic of Germany<br />
[203]. He exhibited exceptional mathematical skills, such as fantastic calculation<br />
abilities, from an early age. His teachers were amazed by his adept<br />
ability to perform complicated mathematical operations, in which he often<br />
outstripped his instructor’s knowledge. While still a student at the Gymnasium<br />
in Lüneberg, he read and absorbed Legendre’s 900 page book on number<br />
theory in six days. In 1846, he enrolled at the University of Göttingen and<br />
took courses from Gauss. In 1847 he moved to the University of Berlin to
Biographical Vignettes 167<br />
study under Steiner, Jacobi, Dirichlet and Eisenstein. In 1849, he returned<br />
to Göttingen and submitted his thesis, supervised by Gauss, in 1851. This<br />
thesis applied topological methods to complex function theory and introduced<br />
Riemann surfaces to study the geometric properties of analytic functions, conformal<br />
mappings and the connectivity of surfaces. (A fundamental theorem on<br />
Riemann surfaces appears in Property 41 of Chapter 2.) In order to become<br />
a Lecturer, he had to work on his Habilitation. In addition to another thesis<br />
(on trigonometric series including a study of Riemann integrability), this required<br />
a public lecture which Gauss chose to be on geometry. The resulting<br />
On the hypotheses that lie at the foundations of geometry of 1854 is considered<br />
a classic of Mathematics. In it, he gave the definition of n-dimensional<br />
Riemannian space and introduced the Riemannian curvature tensor. For the<br />
case of a surface, this reduces to a scalar, the constant non-zero cases corresponding<br />
to the known non-Euclidean geometries. He showed that, in four<br />
dimensions, a collection of ten numbers at each point describe the properties<br />
of a manifold, i.e. a Riemannian metric, no matter how distorted. This provided<br />
the mathematical framework for Einstein’s General Theory of Relativity<br />
sixty years later. This allowed him to begin lecturing at Göttingen, but he<br />
was not appointed Professor until 1857. In 1857, he published another of his<br />
masterpieces, Theory of abelian functions which further developed the idea<br />
of Riemann surfaces and their topological properties. In 1859, he succeeded<br />
Dirichlet as Chair of Mathematics at Göttingen and was elected to the Berlin<br />
Academy of Sciences. A newly elected member was expected to report on<br />
their most recent research and Riemann sent them On the number of primes<br />
less than a given magnitude. This great masterpiece, his only paper on number<br />
theory, introduced the Riemann zeta function and presented a number of<br />
conjectures concerning it, most notably the Riemann Hypothesis, the greatest<br />
unsolved problem in Mathematics [76] (Hilbert’s Eighth Problem [332] and one<br />
of the $1M Millenium Prize Problems [78]) ! It conjectures that, except for<br />
a few trivial exceptions, the roots of the zeta function all have a real part of<br />
1/2 in the complex plane. The Riemann Hypothesis implies results about the<br />
distribution of prime numbers that are in some ways as good as possible. His<br />
work on monodromy and the hypergeometric function in the complex domain<br />
established a basic way of working with functions by consideration of only<br />
their singularities. He died from tuberculosis, aged 39, in Salasca, Italy, where<br />
he was seeking the health benefits of the warmer climate. Source material for<br />
Riemann is available in [23, 42, 287].<br />
Vignette 24 (James Clerk Maxwell: 1831-1879).<br />
James Clerk Maxwell, physicist and Mathematician, was born in Edinburgh,<br />
Scotland [43, 215, 307]. He attended the prestigious Edinburgh Academy<br />
and, at age 14, wrote a paper on ovals where he generalized the definition of an
168 Biographical Vignettes<br />
ellipse by defining the locus of a point where the sum of m times the distance<br />
from one fixed point plus n times the distance from a second fixed point is<br />
constant. (m = n = 1 corresponds to an ellipse.) He also defined curves where<br />
there were more than two foci. This first paper, On the description of oval<br />
curves, and those having a plurality of foci, was read to the Royal Society of<br />
Edinburgh in 1846. At age 16, he entered the University of Edinburgh and,<br />
although he could have attended Cambridge after his first term, he instead<br />
completed the full course of undergraduate studies at Edinburgh. At age 18,<br />
he contributed two papers to the Transactions of the Royal Society of Edinburgh.<br />
In 1850, he moved to Cambridge University, first to Peterhouse and<br />
then to Trinity where he felt his chances for a fellowship were greater. He<br />
was elected to the secret Society of Apostles, was Second Wrangler and tied<br />
for Smith’s Prizeman. He obtained his fellowship and graduated with a degree<br />
in Mathematics in 1854. Immediately after taking his degree, he read<br />
to the Cambridge Philosophical Society the purely mathematical memoir On<br />
the transformation of surfaces by bending. In 1855, he presented Experiments<br />
on colour to the Royal Society of Edinburgh where he laid out the principles<br />
of colour combination based upon his observations of colored spinning tops<br />
(Maxwell discs). (Application 14 concerns the related Maxwell Color Triangle.)<br />
In 1855 and 1856, he read his two part paper On Faraday’s lines of force<br />
to the Cambridge Philosophical Society where he showed that a few simple<br />
mathematical equations could express the behavior of electric and magnetic<br />
fields and their interaction. In 1856, Maxwell took up an appointment at Marishcal<br />
College in Aberdeen. He spent the next two years working on the nature<br />
of Saturn’s rings and, in 1859, he was awarded the Adams Prize of St. John’s<br />
College, Cambridge for his paper On the stability of Saturn’s rings where he<br />
showed that stability could only be achieved if the rings consisted of numerous<br />
small solid particles, an explanation finally confirmed by the Voyager spacecrafts<br />
in the 1980’s! In 1860, he was appointed to the vacant chair of Natural<br />
Philosophy at King’s College in London. He performed his most important experimental<br />
work during the six years that he spent there. He was awarded the<br />
Royal Society’s Rumford medal in 1860 for his work on color which included<br />
the world’s first color photograph, and was elected to the Society in 1861. He<br />
also developed his ideas on the viscosity of gases (Maxwell-Boltzmann kinetic<br />
theory of gases), and proposed the basics of dimensional analysis. This time<br />
is especially known for the advances he made in electromagnetism: electromagnetic<br />
induction, displacement current and the identification of light as an<br />
electromagnetic phenomenon. In 1865, he left King’s College and returned to<br />
his Scottish estate of Glenlair until 1871 when he became the first Cavendish<br />
Professor of Physics at Cambridge. He designed the Cavendish laboratory<br />
and helped set it up. The four partial differential equations now known as<br />
Maxwell’s equations first appeared in fully developed form in A Treatise on
Biographical Vignettes 169<br />
Electricity and Magnetism (1873) although most of this work was done at<br />
Glenlair. It took until 1886 for Heinrich Hertz to produce the electromagnetic<br />
waves mathematically predicted by Maxwell’s equations. Maxwell’s legacy<br />
to us also includes the Maxwell distribution, Maxwell materials, Maxwell’s<br />
theorem, the generalized Maxwell model and Maxwell’s demon. He died of<br />
abdominal cancer at Cambridge, aged 48.<br />
Vignette 25 (Charles Lutwidge Dodgson/Lewis Carroll:1832-1898).<br />
C. L. Dodgson, a.k.a. Lewis Carroll, was born in Daresbury, Cheshire, England<br />
[52]. He matriculated at Christ Church, Oxford, graduating in 1854 and<br />
becoming Master of Arts there three years later. In 1852, while still an undergraduate,<br />
he won a Fellowship (allowing him to live in Christ Church College<br />
provided that the took Holy Orders and remained unmarried, both of which<br />
he did) and, in 1855, he was appointed Lecturer in Mathematics at his alma<br />
mater, where he stayed in various capacities until his death. His mathematical<br />
contributions included Elementary Treatise on Determinants (1867), A Discussion<br />
of the Various Procedures in Conducting Elections (1873), Euclid and<br />
His Modern Rivals (1879), The Game of Logic (1887), Curiosa Mathematica<br />
(1888/1893) and Symbolic Logic (1896). Despite these many and varied publications,<br />
he is best remembered for his children’s stories Alice’s Adventures<br />
in Wonderland (1865) and Through the Looking Glass (1872). Incidentally,<br />
Martin Gardner’s most commercially successful books were his annotations of<br />
these children’s classics. Even more interesting is that he was asked to undertake<br />
this publication venture only after the publishers could not get their<br />
first choice - Bertrand Russell! Dodgson/Carroll’s predilection for paper folding<br />
was alluded to in Recreation 28. He died suddenly from what began as a<br />
minor cold in Guildford, Surrey, England, aged 65.<br />
Vignette 26 (Hermann Amandus Schwarz: 1843-1921).<br />
Hermann Schwarz was born in Hermsdorf, Silesia (now part of Poland)<br />
[144]. Initially, he studied chemistry at the Technical University of Berlin<br />
but switched to Mathematics, receiving his doctorate in 1864 for a thesis in<br />
algebraic geometry written under Weierstrass and examined by Kummer (his<br />
eventual father-in-law). He then taught at University of Halle and ETH-Zurich<br />
until accepting the Chair of Mathematics at Göttingen in 1875. In 1892, he<br />
returned to University of Berlin as Professor of Mathematics. His greatest<br />
strength lay in his geometric intuition as is evidenced by his first publication,<br />
an elementary proof of the chief theorem of axonometry (a method for mapping<br />
three-dimensional images onto the plane). He made important contributions to<br />
conformal mappings and minimal surfaces. His legacy to Mathematics is vast:<br />
Schwarz alternating method, Schwarzian derivative, Schwarz’ lemma, Schwarz<br />
minimal surface, Schwarz-Christoffel formula, Cauchy-Schwarz inequality and
170 Biographical Vignettes<br />
Schwarz reflection priniciple. The latter has previously been described in the<br />
context of Fagnano’s Problem (Property 53) and Triangular Billiards (Recreation<br />
20). He died in Berlin, aged 78. Source material for Schwarz is available<br />
in [23].<br />
Vignette 27 (Jules Henri Poincaré: 1854-1912).<br />
Henri Poincaré, described by many as The Last Universalist in Mathematics,<br />
was born into an upper middle class family in Nancy, France [67]. He<br />
was not the only distinguished member of his family. His cousin, Raymond<br />
Poincaré, was several times Prime Minister of France and President of the<br />
French Republic during World War I. In 1862, Henri entered the Lycée in<br />
Nancy (now renamed after him) and spent eleven years there as one of the top<br />
students in every subject. He won first prizes in the concours général, a competition<br />
between the top students from all across France. In 1873, he entered<br />
l’ École Polytechnique, graduating in 1875. After graduation, he continued his<br />
studies at l’ École des Mines after which he spent a short time working as a<br />
mining engineer while completing his doctoral work. In 1879, he received his<br />
doctorate under Charles Hermite at the University of Paris with a thesis on differential<br />
equations where he introduced the qualitative geometric theory which<br />
was to become so influential. He then was appointed to teach mathematical<br />
analysis at the University of Caen. In 1881, he became a Professor at the<br />
University of Paris and also at l’ École Polytechnique, holding both posts for<br />
the rest of his life. The breadth and depth of his mathematical contributions<br />
is truly staggering. He won a mathematical competition based on his work on<br />
the three-body problem which used invariant integrals, introduced homoclinic<br />
points and gave the first mathematical description of chaotic motion. He also<br />
made fundamental contributions to number theory, automorphic functions and<br />
the theory of analytic functions of several complex variables. His work in algebraic<br />
topology was especially noteworthy where he created homotopy theory<br />
and introduced the notion of the fundamental group as well as formulated the<br />
celebrated Poincaré Conjecture which has only recently been settled in the<br />
affirmative by Grigory Perelman [141]. In Applied Mathematics, he made advances<br />
in fluid mechanics, optics, electricity, telegraphy, capillarity, elasticity,<br />
thermodynamics, potential theory, quantum theory, theory of relativity and<br />
celestial mechanics, the latter culminating in his masterpiece Les Méthodes<br />
nouvelles de la mécanique céleste in three volumes published between 1892<br />
and 1899. See Property 84 of Chapter 2 for a description of the Poincaré disk<br />
model of the hyperbolic plane. His name has been enshrined in the Poincaré-<br />
Bendixson Theorem, the Poincaré Group, the Poincaré-Linstedt Method, the<br />
Poincaré Inequality, the Poincaré Metric and the Poincaré Map, to mention<br />
but a few. Poincaré’s popular works included Science and Hypothesis (1901),<br />
The Value of Science (1905) and Science and Method (1908). He was the only
Biographical Vignettes 171<br />
member of the French Academy of Sciences to be elected to every one of its<br />
five sections and he served as its President. In addition, he received many<br />
medals and honors. He died from complications following prostate surgery in<br />
Paris, France, aged 58. Source material for Poincaré is available in [23, 42].<br />
Vignette 28 (Percy Alexander MacMahon: 1854-1929).<br />
Percy MacMahon was born into a military family in Sliema, Malta [144]. In<br />
1871, he entered the Royal Military Academy at Woolwich and studied under<br />
the renowned teacher of physics and Mathematics, Alfred George Greenhill.<br />
He was posted to India in 1873 until he was sent home to England in 1878 to<br />
recover his health. He was appointed Instructor of Mathematics at the Royal<br />
Military Academy in 1882 and held that post until he became Assistant Inspector<br />
at the Arsenal in 1888. In 1891, he took up a new post as Military<br />
Instructor in Electricity at the Royal Artillery College, Woolwich where he<br />
stayed until his retirement from the Army in 1898. He worked on invariants<br />
of binary quadratic forms and his interest in symmetric functions led him to<br />
study partitions of integers and Latin squares. In 1915/1916, he published his<br />
two volume Combinatory Analysis which was the first major book in enumerative<br />
combinatorics and is now considered a classic. The shorter Introduction to<br />
Combinatory Analysis was published in 1920. He also did pioneering work in<br />
Recreational Mathematics and patented several puzzles. His New Mathematical<br />
Pastimes (1921) contains the 24 color triangles introduced in Recreation<br />
17. He was a Fellow of the Royal Society and served as President of the London<br />
Mathematical Society, Section A of the British Association and the Royal Astronomical<br />
Society. He was also the recipient of the Royal Medal, the Sylvester<br />
Medal and the Morgan Medal. He died in Bognor Regis, England, aged 75.<br />
Vignette 29 (Frank Morley: 1860-1937).<br />
Frank Morley was born into a Quaker family in Woodbridge, Suffolk, England<br />
[230, 330]. He studied with Sir George Airy at King’s College, Cambridge,<br />
earning his B.A. in 1884. He then took a job as a school master, teaching<br />
Mathematics at Bath College until 1887. At that time he moved to Haverford<br />
College in Pennsylvania where he taught until 1900, when he became<br />
Chairman of the Mathematics Department at the Johns Hopkins University in<br />
Baltimore, Maryland. He spent the remainder of his career there, supervising<br />
48 doctoral students. He published the book A Treatise on the Theory of Functions<br />
(1893) which was later revised as Introduction to the Theory of Analytic<br />
Functions (1898). He is best known for Morley’s Theorem (see Property 11),<br />
which though discovered in 1899 was not published by him until 1929, but also<br />
loved posing mathematical problems. Over a period of 50 years, he published<br />
more than 60 such problems in Educational Times. Most were of a geometric<br />
nature: “Show that on a chessboard the number of visible squares is 204 while
172 Biographical Vignettes<br />
the number of visible rectangles (including squares) is 1,296; and that, on a<br />
similar board with n squares on a side, the number of squares is the sum of<br />
the first n square numbers while the number of rectangles (including squares)<br />
is the sum of the first n cube numbers.” He was President of the American<br />
Mathematical Society and Editor of American Journal of Mathematics<br />
(where he finally published Morley’s Theorem). He was also an exceptional<br />
chess player, having defeated fellow Mathematician Emmanuel Lasker while<br />
the latter was still reigning World Champion! His three sons became Rhodes<br />
Scholars: Christopher became a famous novelist, Felix became Editor of The<br />
Washington Post and also President of Haverford College, and Frank became<br />
director of the publishing firm Faber and Faber but was also a Mathematician<br />
who published Inversive Geometry with his father in 1933. He died in<br />
Baltimore, aged 77.<br />
Vignette 30 (Hermann Minkowski: 1864-1909).<br />
Hermann Minkowski was born of German parents in Alexotas, a suburb<br />
of Kaunas, Lithuania which was then part of the Russian Empire [144]. The<br />
family returned to Germany and settled in Königsberg when he was eight years<br />
old. He received his higher education at the University of Königsberg where he<br />
became a lifelong friend of David Hilbert, his fellow student, and Adolf Hurwitz,<br />
his slightly older teacher. In 1883, while still a student at Königsberg,<br />
he was awarded the Mathematics Prize from the French Academy of Sciences<br />
for his manuscript on the theory of quadratic forms. His 1885 doctoral thesis<br />
at Königsberg was a continuation of this prize winning work. In 1887 he<br />
moved to the University of Bonn where he taught until 1894, then he returned<br />
to Königsberg for two years before becoming a colleague of Hurwitz at ETH,<br />
Zurich in 1896 where Einstein was his student. In 1896, he presented his Geometry<br />
of Numbers, a geometrical method for solving problems in number<br />
theory. In 1902, he joined the Mathematics Department of the University of<br />
Göttingen where he was reunited with Hilbert (who had arranged to have the<br />
chair created specifically for Minkowski) and he stayed there for the rest of<br />
his life. It is of great historical interest that it was in fact Minkowski who<br />
suggested to Hilbert the subject of his famous 1900 lecture in Paris on “the<br />
Hilbert Problems” [332]. In 1907, he realized that Einstein’s Special Theory of<br />
Relativity could best be understood in a non-Euclidean four-dimensional space<br />
now called Minkowski spacetime in which time and space are not separate entities<br />
but instead are intermingled. This space-time continuum provided the<br />
framework for all later mathematical work in this area, including Einstein’s<br />
General Theory of Relativity. In 1907, he published his Diophantische Approximationen<br />
which gave an elementary account of his work on the geometry<br />
of numbers and of its application to Diophantine approximation and algebraic<br />
numbers. His subsequent work on the geometry of numbers led him to investigate<br />
convex bodies and packing problems. His Geometrie der Zahlen was
Biographical Vignettes 173<br />
published posthumously in 1910. M-matrices were named for him by Alexander<br />
Ostrowski. See Property 85 of Chapter 2 for a result on the Minkowski<br />
plane with a regular dodecagon as unit circle. He died suddenly of appendicitis<br />
in Göttingen, aged 44.<br />
Vignette 31 (Helge von Koch: 1870-1924).<br />
Helge von Koch was born into a family of Swedish nobility in Stockholm<br />
[144]. In 1892, he earned his doctorate under Gösta Mittag-Leffler at Stockholm<br />
University. Between the years 1893 and 1905, von Koch had several<br />
appointments as Assistant Professor of Mathematics until he was appointed to<br />
the Chair of Pure Mathematics at the Royal Institute of Technology in 1905,<br />
succeeding Ivar Bendixson. In 1911, he succeeded Mittag-Leffler as Professor<br />
of Mathematics at Stockholm University. Von Koch is known principally for<br />
his work in the theory of infinitely many linear equations and the study of the<br />
matrices derived from such infinite systems. He also did work in differential<br />
equations and the theory of numbers. One of his results was a 1901 theorem<br />
proving that the Riemann Hypothesis is equivalent to a stronger form of the<br />
Prime Number Theorem. He invented the Koch Snowflake (see Propery 60) in<br />
his 1904 paper titled “On a continuous curve without tangents constructible<br />
from elementary geometry”. He died in Stockholm, aged 54.<br />
Vignette 32 (Bertrand Russell: 1872-1970).<br />
Bertrand Russell, 3rd Earl Russell, was born into a liberal family of the<br />
British aristocracy in Trelleck, Monmouthshire, Wales [51]. Due to the death<br />
of his parents, he was raised by his paternal grandparents. He was educated<br />
at home by a series of tutors before entering Trinity College, Cambridge as a<br />
scholar in 1890. There, he was elected to the Apostles where he met Alfred<br />
North Whitehead, then a mathematical lecturer at Cambridge. He earned<br />
his B.A. in 1893 and added a fellowship in 1895 for his thesis, An Essay on<br />
the Foundations of Geometry, which was published in 1897. Despite his previously<br />
noted criticism of The Elements (see opening paragraph of Chapter<br />
1), it was his exposure to Euclid through his older brother Frank that set his<br />
life’s path of work in Mathematical Logic! Over a long and varied career, he<br />
made ground-breaking contributions to the foundations of Mathematics, the<br />
development of formal logic, as well as to analytic philosophy. His mathematical<br />
contributions include the discovery of Russell’s Paradox, the development<br />
of logicism (i.e. that Mathematics is reducible to formal logic), introduction<br />
of the theory of types and the refinement of the first-order predicate calculus.<br />
His other mathematical publications include Principles of Mathematics (1903),<br />
Principia Mathematica with Whitehead (1910, 1912, 1913) and Introduction<br />
to Mathematical Philosophy (1919). Although elected to the Royal Society in<br />
1908, he was convicted and fined in 1916 for his anti-war activities and, as
174 Biographical Vignettes<br />
a consequence, dismissed from Trinity. Two years later, he was convicted a<br />
second time and served six months in prison (where he wrote Introduction to<br />
Mathematical Philosophy). He did not return to Trinity until 1944. He was<br />
married four times and was notorious for his many affairs. Together with his<br />
second wife, he opened and ran an experimental school during the late 1920’s<br />
and early 1930’s. He became the third Earl Russell upon the death of his<br />
brother in 1931. While teaching in the United States in the late 1930’s, he was<br />
offered a teaching appointment at City College of New York but the appointment<br />
was revoked following a large number of public protests and a judicial<br />
decision in 1940 which stated that he was morally unfit to teach youth. He<br />
was awarded the Order of Merit in 1949 and the Nobel Prize for Literature in<br />
1950. In 1961, he was once again imprisoned in connection with anti-nuclear<br />
protests. He died in Penrhyndeudraeth, Merioneth, Wales, aged 97.<br />
Vignette 33 (Henri Lebesgue: 1875-1941).<br />
Henri Lebesgue was born in Beauvais, France and studied at l’ École Normale<br />
Supérieure from 1894 to 1897, at which time he was awarded his teaching<br />
diploma in Mathematics [144]. He spent the next two years working in its library<br />
studying the works of René Baire on discontinuous functions. In 1898,<br />
he published his first paper on polynomial approximation where he introduced<br />
the Lebesgue constant. From 1899 to 1902, while teaching at the Lycée Centrale<br />
in Nancy, he developed the ideas that he presented in 1902 as his doctoral<br />
thesis, “Intégrale, longueur, aire”, written under the supervision of Émile Borel<br />
at the Sorbonne. This thesis, considered to be one of the finest ever written by<br />
a Mathematician, introduced the pivotal concepts of Lebesgue measure and<br />
the Lebesgue integral. He then taught at Rennes (1902-1906) and Poitiers<br />
(1906-1910) before returning to the Sorbonne in 1910. In 1921, he was named<br />
Professor of Mathematics at the Collège de France, a position he held until his<br />
death. He is also remembered for the Riemann-Lebesgue lemma, Lebesgue’s<br />
dominated convergence theorem, the Lebesgue-Stieltjes integral, the Lebesgue<br />
number and Lebesgue covering dimension in topology and the Lebesgue spine<br />
in potential theory. Property 56 of Chapter 2 contains a statement of the<br />
Blaschke-Lebesgue Theorem on curves of constant breadth. He was a member<br />
of the French Academy of Sciences, the Royal Society, the Royal Academy of<br />
Science and Letters of Belgium, the Academy of Bologna, the Accademia dei<br />
Lincei, the Royal Danish Academy of Sciences, the Romanian Academy and<br />
the Kraków Academy of Science and Letters. He was a recipient of the Prix<br />
Houllevigue, the Prix Poncelet, the Prix Saintour and the Prix Petit d’Ormoy.<br />
He died in Paris, France, aged 66.<br />
Vignette 34 (Waclaw Sierpinski: 1882-1969).<br />
Waclaw Sierpinski was born in Warsaw which at that time was part of the<br />
Russian Empire [144]. He enrolled in the Department of Mathematics and
Biographical Vignettes 175<br />
Physics of the University of Warsaw. After his graduation in 1904, he worked<br />
as a school teacher teacher in Warsaw before enrolling for graduate study at<br />
the Jagiellonian University in Kraków. He received his doctorate in 1906 under<br />
S. Zaremba and G. F. Voronoi and was appointed to the University of Lvov<br />
in 1908. He spent the years of World War I in Moscow working with Nikolai<br />
Luzin and returned to Lvov afterwards. Shortly thereafter, he accepted a post<br />
at the University of Warsaw where he spent the rest of his life. He made many<br />
outstanding contributions to set theory, number theory, theory of functions<br />
and topology. He published over 700 papers and 50 books. Three well-known<br />
fractals are named after him: the Sierpinski gasket (see Properties 60-62 and<br />
Application 24), the Sierpinski carpet and the Sierpinski curve. In number<br />
theory, a Sierpinski number is an odd natural number k such that all integers<br />
of the form k · 2 n + 1 are composite for all natural numbers n. In 1960, he<br />
proved that there are infinitely many such numbers and the Sierpinski Problem,<br />
which is still open to this day, is to find the smallest one. He was intimately<br />
involved in the development of Mathematics in Poland, serving as Dean of the<br />
Faculty at the University of Warsaw and Chairman of the Polish Mathematical<br />
Society. He was a founder of the influential mathematical journal Fundamenta<br />
Mathematica and Editor-in-Chief of Acta Arithmetica. He was a member of<br />
the Bulgarian Academy of Sciences, the Accademia dei Lincei of Rome, the<br />
German Academy of Sciences, the U.S. National Academy of Sciences, the<br />
Paris Academy, the Royal Dutch Academy, the Romanian Academy and the<br />
Papal Academy of Sciences. In 1949 he was awarded Poland’s Scientific Prize,<br />
First Class. He died in Warsaw, Poland, aged 87.<br />
Vignette 35 (Wilhelm Blaschke: 1885-1962).<br />
Wilhelm Blaschke was born in Graz, Austria, son of Professor of Descriptive<br />
Geometry Josef Blaschke [144]. Through his father’s influence, he became a<br />
devotee of Steiner’s concrete geometric approach to Mathematics. He studied<br />
architectural engineering for two years at the Technische Hochschule in Graz<br />
before going to the Universtiy of Vienna where he earned his doctorate under<br />
W. Wirtinger in 1908. He then visited different universities (Pisa, Göttingen,<br />
Bonn, Breifswald) to study with the leading geometers of the day. He next<br />
spent two years at Prague and two more years at Leipzig where he published<br />
Kreis und Kugel (1916) in which he investigated isoperimetric properties of<br />
convex figures in the style of Steiner. He then went to Königsberg for two<br />
years, briefly went to Tübingen, until finally being appointed to a chair at the<br />
University of Hamburg where he stayed (with frequent visits to universities<br />
around the world) for the remainder of his career. At Hamburg, he built an<br />
impressive department by hiring Hecke, Artin and Hasse. During World War<br />
II, he joined the Nazi Party, a decision that was to haunt him afterwards.<br />
He wrote an important book, Vorlesungen über Differentialgeometrie (1921-<br />
1929), which was a major three volume work. He also initiated the study of
176 Biographical Vignettes<br />
topological differential geometry. See Property 27 for Blaschke’s Theorem and<br />
Property 56 for the Blaschke-Lebesgue Theorem. He died in Hamburg, aged<br />
76.<br />
Vignette 36 (Richard Buckminster Fuller, Jr.: 1895-1983).<br />
R. Buckminster Fuller was a famed architect, engineer and Mathematician<br />
born in Milton, Massachusetts [279]. Bucky, as he was known, holds the dubious<br />
distinction of having been expelled from Harvard - twice! Business disasters<br />
and the death of his four year old daughter brought him to the brink of suicide,<br />
but instead he shifted the course of his life to showing that technology could<br />
be beneficial to mankind if properly used. He developed a vectorial system of<br />
geometry, Synergetics [113], based upon the tetrahedron which provides maximum<br />
strength with minimum structure. He coined the term Spaceship Earth<br />
to emphasize his belief that we must work together globally as a crew if we are<br />
to survive. He is best known for his Dymaxion House, Dymaxion Car, Dymaxion<br />
Map (see Application 29) and geodesic dome (see Application 30). More<br />
than 200, 000 of the latter have been built, the most famous being the United<br />
States Pavillion at the 1967 International Exhibition in Montreal. He has been<br />
immortalized in fullerenes which are molecules composed entirely of carbon in<br />
the form of a hollow sphere (buckyball), ellipsoid or tube. Specifically, C60 was<br />
the first to be discovered and is named buckminsterfullerene. He eventually<br />
became a Professor at Southern Illinois University until his retirement in 1975.<br />
He died in Los Angeles, California, aged 87.<br />
Vignette 37 (Harold Scott MacDonald Coxeter: 1907-2003).<br />
Donald Coxeter was born in London and educated at University of Cambridge<br />
[254]. He received his B.A. in 1929 and his doctorate in 1931 with the<br />
thesis Some Contributions to the Theory of Regular Polytopes written under<br />
the supervision of H. F. Baker. He then became a Fellow at Cambridge and<br />
spent two years as a research visitor at Princeton University. He then joined<br />
the faculty at University of Toronto in 1936 where he stayed for the remaining<br />
67 years of his life. His research was focused on geometry where he made major<br />
contributions to the theory of polytopes (Coxeter polytopes), non-Euclidean<br />
geometry, group theory (Coxeter groups) and combinatorics. In 1938, he revised<br />
and updated Rouse Ball’s Mathematical Recreations and Essays, first<br />
published in 1892 and still widely read today. He wrote a number of widely<br />
cited geometry books including The Real Projective Plane (1955), Introduction<br />
to Geometry (1961), Regular Polytopes (1963), Non-Euclidean Geometry<br />
(1965) and Geometry Revisited (1967). He also published 167 research articles.<br />
He was deeply interested in music and art: at one point he pondered becoming<br />
a composer and was a close friend of M. C. Escher. Another of his friends,<br />
R. Buckminster Fuller utilized his geometric ideas in his architecture. His role
Biographical Vignettes 177<br />
in popularizing the mathematical work of the institutionalized artist George<br />
Odom has already been described in Chapter 1 and Property 74. He was a<br />
Fellow of the Royal Societies of London and Canada as well as a Companion<br />
of the Order of Canada, their highest honor. He died in Toronto, aged 96, and<br />
attributed his longevity to strict vegetarianism as well as an exercise regimen<br />
which included 50 daily push-ups.<br />
Vignette 38 (Paul Erdös: 1913-1996).<br />
Paul Erdös was born in Budapest, Hungary to Jewish parents both of<br />
whom were Mathematics teachers [175, 269]. His fascination with Mathematics<br />
developed early as is evidenced by his ability, at age three, to calculate<br />
how many seconds a person had lived. In 1934, at age 21, he was awarded a<br />
doctorate in Mathematics from Eötvös Loránd University for the thesis Über<br />
die Primzahlen gewisser arithmetischer Reihen written under the supervision<br />
of L. Fejér. Due to a rising tide of anti-Semitism, he immediately accepted<br />
the position of Guest Lecturer in Mathematics at Manchester University in<br />
England and, in 1938, he accepted a Fellowship at Princeton University. He<br />
then held a number of part-time and temporary positions which eventually led<br />
to an itinerant existence. To describe Erdös as peripatetic would be to risk<br />
the mother of all understatements. He spent most of his adult life living out<br />
of a single suitcase (sometimes traveling with his mother), had no checking<br />
account and rarely stayed consecutively in one place for more than a month.<br />
Friends and collaborators such as Ron Graham (see below) helped him with<br />
the mundane details of modern life. He was so eccentric that even his close<br />
friend, Stan Ulam, described him thusly: “His peculiarities are so numerous<br />
that it is impossible to describe them all.” He had his own idiosyncratic vocabulary<br />
including The Book which referred to an imaginary book in which<br />
God (whom he called the “Supreme Fascist”) had written down the most elegant<br />
proofs of mathematical theorems. He foreswore any sexual relations and<br />
regularly abused amphetamines. Mathematically, he was a problem solver and<br />
not a theory builder, frequently offering cash prizes for solutions to his favorite<br />
problems. He worked primarily on problems in combinatorics, graph theory,<br />
number theory, classical analysis, approximation theory, set theory and probability<br />
theory. His most famous result is the discovery, along with Atle Selberg,<br />
of an elementary proof of the Prime Number Theorem. See Properties 48 and<br />
55 and Recreation 13 for a further discussion of his contributions. He published<br />
more papers (approximately 1475) than any other Mathematician in history<br />
with 511 coauthors. (Euler published more pages.) This prolific output led to<br />
the concept of the Erdös number which measures the collaborative distance<br />
between him and other Mathematicians. He was the recipient of the Cole<br />
and Wolf Prizes and was an Honorary Member of the London Mathematical<br />
Society. He died, aged 86, while attending, naturally enough, a Mathematics<br />
conference in Warsaw, Poland.
178 Biographical Vignettes<br />
Vignette 39 (Solomon Wolf Golomb: 1932-).<br />
Solomon W. Golomb, Mathematician and engineer, was born in Baltimore,<br />
Maryland [330]. He received his B.A. from Johns Hopkins in 1951 and his<br />
M.A. (1953) and Ph.D. (1957) in Mathematics from Harvard, where he wrote<br />
the thesis Problems in the Distribution of Prime Numbers under the supervision<br />
of D. V. Widder. He has worked at the Glenn L. Martin Company, where<br />
he became interested in communications theory and began working on shift<br />
register sequences, and the Jet Propulsion Laboratory at Caltech. In 1963,<br />
he joined the faculty of University of Southern California, where he remains<br />
today, with a joint appointment in the Departments of Electrical Engineering<br />
and Mathematics. His research has been specialized in combinatorial analysis,<br />
number theory, coding theory and communications. Today, millions of cordless<br />
and cellular phones rely upon his fundamental work on shift register sequences.<br />
However, he is best known as the inventor of Polyominoes (1953), the inspiration<br />
for the computer game Tetris. His other contributions to Recreational<br />
Mathematics include the theory of Rep-tiles (Recreation 15) and Hexiamonds<br />
(Recreation 16). He has been a regular columnist in Scientific American, IEEE<br />
Information Society Newsletter and Johns Hopkins Magazine. He has been the<br />
recipient of the NSA Research Medal, the Lomonosov and Kapitsa Medals of<br />
the Russian Academy of Sciences and the Richard W. Hamming Medal of the<br />
IEEE. He is also a Fellow of both IEEE and AAAS as well as a member of the<br />
National Academy of Engineering.<br />
Vignette 40 (Ronald Lewis Graham: 1935-).<br />
Ron Graham was born in Taft, California and spent his childhood moving<br />
back and forth between there and Georgia, eventually settling in Florida<br />
[230]. He then entered University of Chicago on a three year Ford Foundation<br />
scholarship at age 15 without graduating from high school. It is here that he<br />
learnt gymnastics and became proficient at juggling and the trampoline. Without<br />
graduating, he spent the next year at University of California at Berkeley<br />
studying electrical engineering before enlisting for four years in the Air Force.<br />
During these years of service, he earned a B.S. in Physics from University of<br />
Alaska. After his discharge, he returned to UC-Berkeley where he completed<br />
his Ph.D. under D.H. Lehmer in 1962 with the thesis On Finite Sums of Rational<br />
Numbers. He then joined the technical staff of Bell Telephone Laboratories<br />
where he worked on problems in Discrete Mathematics, specifically scheduling<br />
theory, computational geometry, Ramsey theory and quasi-randomness.<br />
(Here, he became Bell Labs and New Jersey ping-pong champion.) In 1963,<br />
he began his long collaboration with Paul Erdös which eventually led to 30<br />
joint publications and his invention of the “Erdös number”. His contributions<br />
to partitioning an equilateral triangle have been described in Property 80 of<br />
Chapter 2. In 1977, he entered the Guinness Book of Records for what is now
Biographical Vignettes 179<br />
known as Graham’s number, the largest number ever used in a mathematical<br />
proof. He has also appeared in Ripley’s Believe It or Not for not only being one<br />
of the world’s foremost Mathematicians but also a highly skilled trampolinist<br />
and juggler. In fact, he has served as President of the American Mathematical<br />
Society, Mathematical Association of America and the International Jugglers’<br />
Association!. In 1999, he left his position as Director of Information Sciences at<br />
Bell Labs to accept a Chaired Professorship at University of California at San<br />
Diego which he still holds. He has been the recipient of the Pólya Prize, the<br />
Allendoerfer Award, the Lester R. Ford Award, the Euler Medal and the Steele<br />
Award. He is a member of the National Academy of Sciences, the American<br />
Academy of Arts and Sciences, the Hungarian Academy of Sciences, Fellow of<br />
the Association of Computing Machinery and the recipient of numerous honorary<br />
degrees. He has published approximately 320 papers (77 of which are<br />
coauthored with his wife, Fan Chung) and five books.<br />
Vignette 41 (John Horton Conway: 1937-).<br />
John Conway, perhaps the world’s most untidy Mathematician, was born in<br />
Liverpool and educated at Gonville and Caius College, Cambridge [230]. After<br />
completing his B.A. in 1959, he commenced research in number theory under<br />
the guidance of Harold Davenport. During his studies at Cambridge, he developed<br />
his interest in games and spent hours playing backgammon in the common<br />
room. He earned his doctorate in 1964, was appointed Lecturer in Pure Mathematics<br />
at Cambridge and began working in mathematical logic. However, his<br />
first major result came in finite group theory when, in 1968, he unearthed a previously<br />
undiscovered finite simple group, of order 8, 315, 553, 613, 086, 720, 000<br />
with many interesting subgroups, in his study of the Leech lattice of sphere<br />
packing in 24 dimensions! He became widely known outside of Mathematics<br />
proper with the appearance of Martin Gardner’s October 1970 Scientific<br />
American article describing his Game of Life. It has been claimed that, since<br />
that time, more computer time has been devoted to it than to any other single<br />
activity. More importantly, it opened up the new mathematical field of<br />
cellular automata. Also in 1970, he was elected to a fellowship at Gonville<br />
and Caius and, three years later, he was promoted from Lecturer to Reader in<br />
Pure Mathematics and Mathematical Statistics at Cambridge. In his analysis<br />
of the game of Go, he discovered a new system of numbers, the surreal numbers.<br />
He has also analyzed many other puzzles and games such as the Soma<br />
cube and peg solitaire and invented many others such as Conway’s Soldiers<br />
and the Angels and Devils Game. He is the inventor of the Doomsday algorithm<br />
for calculating the day of the week and, with S. B. Kochen, proved the<br />
Free Will Theorem of Quantum Mechanics whereby “If experimenters have free<br />
will then so do elementary particles.” A better appreciation of the wide swath<br />
cut by his mathematical contributions can be gained by perusing Property 22
180 Biographical Vignettes<br />
and Recreation 23. In 1983, he was appointed Professor of Mathematics at<br />
Cambridge and, in 1986, he left Cambridge to accept the John von Neumann<br />
Chair of Mathematics at Princeton which he currently holds. He has been<br />
the recipient of the Berwick Prize, the Pólya Prize, the Nemmers Prize, the<br />
Steele Prize and is a Fellow of the Royal Society. His many remarkable books<br />
include Winning Ways for Your Mathematical Plays, The Book of Numbers<br />
and The Symmetries of Things. The mathematical world anxiously awaits his<br />
forthcoming The Triangle Book which reportedly will be shaped like a triangle<br />
and will provide the definitive treatment of all things triangular!<br />
Vignette 42 (Donald Ervin Knuth: 1938-).<br />
Donald Knuth was born in Milwaukee, Wisconsin and was originally attracted<br />
more to Music than to Mathematics [230]. (He plays the organ, saxophone<br />
and tuba.) His first brush with notoriety came in high school when he<br />
entered a contest with the aim of finding how many words could be formed<br />
from “Ziegler’s Giant Bar”. He won top prize by forming 4500 words (without<br />
using the apostrophe)! He earned a scholarship to Case Institute of Technology<br />
to study physics but switched to Mathematics after one year. While at<br />
Case, he was hired to write compilers for various computers and wrote a computer<br />
program to evaluate the performance of the basketball team which he<br />
managed. This latter activity garnered him press coverage by both Newsweek<br />
and Walter Cronkite’s CBS Evening News. He earned his B.S. in 1960 and,<br />
in a unique gesture, Case awarded him an M.S. at the same time. That same<br />
year, he published his first two papers. He then moved on to graduate study<br />
at California Institute of Technology where he was awarded his Ph.D. in 1963<br />
for his thesis Finite Semifields and Projective Planes written under the supervision<br />
of Marshall Hall, Jr. While still a doctoral candidate, Addison-Wesley<br />
approached him about writing a text on compilers which eventually grew to<br />
become the multi-volume mammoth The Art of Computer Programming. In<br />
1963, he became an Assistant Professor of Mathematics at Caltech and was<br />
promoted to Associate Professor in 1966. It is during this period that he published<br />
his insightful analysis of triangular billiards that was described in detail<br />
in Recreation 20 of Chapter 4. In 1968, he was appointed Professor of Computer<br />
Science at Stanford University, where he is today Professor Emeritus.<br />
In 1974, he published the mathematical novelette Surreal Numbers describing<br />
Conway’s set theory construction of an alternative system of numbers.<br />
Starting in 1976, he took a ten year hiatus and invented TeX, a language for<br />
typesetting mathematics, and METAFONT, a computer system for alphabet<br />
design. These two contributions have literally revolutionized the field of scientific<br />
publication. He is also widely recognized as the Father of Analysis of<br />
Algorithms. He has been the recipient of the Grace Murray Hopper Award,<br />
the Alan M. Turing Award, the Lester R. Ford Award, the IEEE Computer
Biographical Vignettes 181<br />
Pioneer Award, the National Medal of Science, the Steele Prize, the Franklin<br />
Medal, the Adelskold Medal, the John von Neumann Medal, the Kyoto Prize,<br />
the Harvey Prize and the Katayanagi Prize. He is a Fellow of the American<br />
Academy of Arts and Science and a member of the National Academy of<br />
Sciences, the National Academy of Engineering, Académie des Sciences, the<br />
Royal Society of London and the Russian Academy of Sciences, as well as an<br />
Honorary Member of the IEEE. In 1990, he gave up his e-mail address so that<br />
he might concentrate more fully on his work and, since 2006, he has waged a<br />
(thus far) successful battle against prostate cancer.<br />
Vignette 43 (Samuel Loyd, Sr.: 1841-1911).<br />
Sam Loyd has been described by Martin Gardner as “America’s greatest<br />
puzzlist and an authentic American genius” [119]. His most famous work is<br />
Cyclopedia of Puzzles (1914) [210] which was published posthumously by his<br />
son. The more mathematical puzzles from this magnum opus were selected<br />
and edited by Martin Gardner [211, 212]. He was born in Philadelphia and<br />
raised in Brooklyn, New York. Rather than attending college, he supported<br />
himself by composing and publishing chess problems. At age 16, he became<br />
problem editor of Chess Monthly and later wrote a weekly chess page for Scientific<br />
American Supplement. (Many of his contributions appeared under such<br />
monikers as W. King, A. Knight and W. K. Bishop.) Most of these columns<br />
were collected in his book Chess Strategy (1878). In 1987, he was inducted<br />
into the U.S. Chess Hall of Fame for his chess compositions. After 1870, the<br />
focus of his work shifted toward mathematical puzzles, some of which were<br />
published in newspapers and magazines while others were manufactured and<br />
marketed. His Greek Symbol Puzzle is considered in Recreation 1 of Chapter<br />
4. He died at his home on Halsey Street in Brooklyn, aged 70.<br />
Vignette 44 (Henry Ernest Dudeney: 1857-1930).<br />
Henry Ernest Dudeney has been described by Martin Gardner as “England’s<br />
greatest inventor of puzzles; indeed, he may well have been the greatest<br />
puzzlist who ever lived” [120]. His most famous works are The Canterbury<br />
Puzzles (1907) [85], Amusements in Mathematics (1917) [86], Modern Puzzles<br />
(1926) and Puzzles and Curious Problems (1931). The last two were combined<br />
and edited by Martin Gardner [87]. He was born in the English village<br />
of Mayfield, East Sussex and, like Loyd, entered a life of puzzling through a<br />
fascination with chess problems. His lifelong involvement with puzzles (often<br />
published in newspapers and magazines under the pseudonym of “Sphinx”)<br />
was done against the backdrop of a career in the Civil Service. For twenty<br />
years, he wrote the successful column “Perplexities” in The Strand magazine<br />
(of Sherlock Holmes fame!). For a time, he engaged in an active correspondence<br />
with Loyd (they even collaborated on a series of articles without ever
182 Biographical Vignettes<br />
meeting) but broke it off, accusing Loyd of plagiarism. His hobbies, other<br />
than puzzling, included billiards, bowling and, especially, croquet, and he was<br />
a skilled pianist and organist. A selection of his puzzles are considered in<br />
Recreations 2-5 of Chapter 4. He died at his home in Lewes, Sussex, aged 73.<br />
Vignette 45 (Martin Gardner: 1914-2010).<br />
For 25 years Martin Gardner wrote “Mathematical Games and Recreations”,<br />
a monthly column for Scientific American magazine. (An anthology<br />
of these columns is available in [137].) He was the author of more than 70<br />
books, the vast majority of which deal with mathematical topics. He was born<br />
and grew up in Tulsa, Oklahoma. He earned a degree in philosophy from University<br />
of Chicago and also began graduate studies there. He served in the<br />
U.S. Navy during World War II as ship’s secretary aboard the destroyer escort<br />
USS Pope. For many years, he lived in Hastings-on-Hudson, New York (on<br />
Euclid Avenue!) and earned his living as a freelance writer, although in the<br />
early 1950’s he was editor of Humpty Dumpty Magazine. In 1979, he semiretired<br />
and moved to Henderson, North Carolina. In 2002, he returned home<br />
to Norman, Oklahoma. Some of his more notable contributions to Recreational<br />
Mathematics are discussed in Property 22 of Chapter 2 and Recreation 20 of<br />
Chapter 4. Despite not being a “professional” mathematician, the American<br />
Mathematical Society awarded him the Steele Prize in 1987, in recognition of<br />
the generations of mathematicians inspired by his writings. The Mathematical<br />
Association of America has honored him for his contributions by holding<br />
a special session on Mathematics related to his work at its annual meeting in<br />
1982 and by making him an Honorary Member of the Association. He was<br />
also an amateur magician thereby making him a bona fide Mathemagician!<br />
He died at a retirement home in Norman, aged 95.
Biographical Vignettes 183<br />
Figure 6.2: Pythagoras Figure 6.3: Plato Figure 6.4: Euclid<br />
Figure 6.5: Archimedes Figure 6.6: Apollonius Figure 6.7: Pappus<br />
Figure 6.8: Fibonacci Figure 6.9: Da Vinci Figure 6.10: Tartaglia
184 Biographical Vignettes<br />
Figure 6.11: Kepler Figure 6.12: Descartes Figure 6.13: Fermat<br />
Figure 6.14: Torricelli Figure 6.15: Viviani Figure 6.16: Pascal<br />
Figure 6.17: Newton Figure 6.18: Euler Figure 6.19: Malfatti
Biographical Vignettes 185<br />
Figure 6.20: Lagrange Figure 6.21: Gauss Figure 6.22: Steiner<br />
Figure 6.23: Bertrand Figure 6.24: Riemann Figure 6.25: Maxwell<br />
Figure 6.26: Dodgson Figure 6.27: Schwarz Figure 6.28: Poincaré
186 Biographical Vignettes<br />
Figure 6.29: MacMahon Figure 6.30: Morley Figure 6.31: Minkowski<br />
Figure 6.32: Von Koch Figure 6.33: Russell Figure 6.34: Lebesgue<br />
Figure 6.35: Sierpinski Figure 6.36: Blaschke Figure 6.37: Fuller
Biographical Vignettes 187<br />
Figure 6.38: Coxeter Figure 6.39: Erdös Figure 6.40: Golomb<br />
Figure 6.41: Graham Figure 6.42: Conway Figure 6.43: Knuth<br />
Figure 6.44: Loyd Figure 6.45: Dudeney Figure 6.46: Gardner
Appendix A<br />
Gallery of Equilateral Triangles<br />
Equilateral triangles appear throughout the Natural and Man-made worlds.<br />
This appendix includes a pictorial panorama of such equilateral delicacies.<br />
Figure A.1: Equilateral Triangular (ET) Humor<br />
188
Gallery 189<br />
Figure A.2: ET Cat Figure A.3: ET Ducks<br />
Figure A.4: ET Geese Figure A.5: ET Fighters<br />
Figure A.6: ET Tulip Figure A.7: ET Bombers
190 Gallery<br />
Figure A.8: ET Wing<br />
Figure A.9: ET Moth Figure A.10: ET UFO<br />
Figure A.11: Winter ET Figure A.12: Tahiti ET
Gallery 191<br />
Figure A.13: ET Lunar Crater<br />
Figure A.14: ET Rock Formation Figure A.15: ET Stone<br />
Figure A.16: ET Gem [21] Figure A.17: ET Crystals [163]
192 Gallery<br />
Figure A.18: ET Bowling Figure A.19: ET Pool Balls<br />
Figure A.20: ET Die Figure A.21: ET Game<br />
Figure A.22: Musical ET Figure A.23: ET Philately
Gallery 193<br />
Figure A.24: Sweet ETs Figure A.25: ET Hat<br />
Figure A.26: ET Flag (Philippines) Figure A.27: Scary ETs
194 Gallery<br />
Figure A.28: ET Escher [268] Figure A.29: ET Möbius Band [240]<br />
Figure A.30: ET Shawl [18] Figure A.31: ET Quilt [225]
Gallery 195<br />
Figure A.32: ET Playscape Figure A.33: ET Dome<br />
Figure A.34: ET Chair Figure A.35: ET Danger
196 Gallery<br />
Figure A.36: ET Street Signs Figure A.37: ET Fallout Shelter<br />
Figure A.38: Impossible ET Figure A.39: ET Window
Gallery 197<br />
Figure A.40: Sacred ETs Figure A.41: Secular ETs<br />
Figure A.42: ET House Figure A.43: ET Lodge [68]
198 Gallery<br />
Figure A.44: ET Tragedy (Triangle Waist Co., New York: March 25, 1911)
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Index<br />
Abel, Niels 165<br />
abutting equilateral triangles 47<br />
Acta Arithmetica 175<br />
Adams Prize 168<br />
ad triangulum 18<br />
Aer 9<br />
affine geometry 51<br />
air-line 86<br />
Airy, Sir George 171<br />
Albanifriedhof cemetery 165<br />
alchemical symbols 24<br />
alchemy v, 21, 24, 160-161<br />
Alice’s Adventures in Wonderland /<br />
Through the Looking Glass<br />
169<br />
alleles 90<br />
All-Union Russian Olympiad 139<br />
altar (triangular) 4<br />
American Journal of Mathematics<br />
172<br />
American Mathematical Monthly 69<br />
American Mathematical Society 172,<br />
179, 182<br />
American Mathematics Competition<br />
(AMC) 129-136<br />
Amusements in Mathematics 181<br />
analysis of algorithms 180<br />
Anaximander 149<br />
Angels and Devils Game 169<br />
antenna design vi, 79-81<br />
anticevian corner triangles 39<br />
antipedal triangle 36, 39<br />
Apollonius circles 38, 152<br />
Apollonius of Perga 150-152, 158,<br />
165, 183<br />
221<br />
Apollonius points 38, 152<br />
Appell, Paul 166<br />
applications vi, 78-102<br />
Applied Mathematics v-vi, 101<br />
Aqua 9<br />
Archbishop’s Coat of Arms 10<br />
Archimedean solid 13, 151, 155<br />
Archimedean spiral 152<br />
Archimedes Codex 151<br />
Archimedes of Syracuse 13, 150-151,<br />
154, 158, 183<br />
Archimedes screw 151<br />
ARC triangle 24<br />
areal coordinates 33<br />
Aristotle 149<br />
Arithmetica Infinitorum 76<br />
Ars Magna 154<br />
Artin, Emil 175<br />
Art of Computer Programming 180<br />
Asshur-izir-pal 4<br />
Astronomia Nova 155<br />
Athene 10<br />
Aubrey, John 148<br />
augmented triad 25<br />
Aum 5<br />
Austrian-Polish Mathematics Competition<br />
138, 141<br />
Babylonian clay tablet 3, 4<br />
Babylonian cuneiform 4<br />
Babylonian cylinder 3, 4<br />
Babylon 5 27-28<br />
Baire, René 174<br />
Baker, H. F. 176<br />
Ball, W. W. Rouse 176
222 Index<br />
Barbier’s Theorem 56<br />
barrel sharing problems 121-123<br />
Barrow, Isaac 160<br />
barycentric coordinates 33<br />
Basel Problem 161<br />
basic properties 29-30<br />
Bath College 171<br />
Baxter, A. M. 120<br />
Beeckman, Isaac 156<br />
Beeson, M. J. 61<br />
Beethoven, Ludwig van 25<br />
Belarusan Mathematical Olympiad<br />
140<br />
Bell Telephone Laboratories 178<br />
Bent Pyramid (Dashur) 2<br />
Berlin Academy 161<br />
Bernoulli, Daniel 161<br />
Bernoulli, Johann 161<br />
Bertrand, Joseph 166, 185<br />
Bertrand’s Paradox 109, 166<br />
Bessel, F. W. 164<br />
Biefeld-Brown effect 84<br />
billiard ball packing 94<br />
billiard ball problem 118<br />
bination 69<br />
binomial coefficients 59<br />
binomial theorem 160<br />
Biographical Dictionary of Mathematicians<br />
vi<br />
biographical vignettes vi, 148-187<br />
biomimetic antenna (“Bat’s Ear”)<br />
80-81<br />
black hole 84<br />
Blaschke, Josef 175<br />
Blaschke-Lebesgue Theorem 56, 174,<br />
176<br />
Blaschke’s Theorem 45, 176<br />
Blaschke, Wilhelm 45, 175, 186<br />
Blundon’s inequality v, 53<br />
Bodleian Library 150<br />
bombers (triangular) 189<br />
The Book 177<br />
Borda voting 91<br />
Borel, Émile 174<br />
Borelli, Giovanni Alfonso 158<br />
Borromean rings 26<br />
bowling pins (triangular) 192<br />
Brahe, Tycho 155<br />
Brahma 5<br />
Brahms, Johannes 25<br />
Brief Lives 148<br />
Briggs, Henry 148<br />
Brotherhood of the Rosy Cross 21<br />
Brown, T. T. 84<br />
Brunnian link 5<br />
Buchman, E. 70<br />
buckminsterfullerene 176<br />
buckyball 176<br />
Budapest University of Art and Design<br />
128<br />
Buddha 6<br />
Buddhism v<br />
building (triangular) 197<br />
Bulgarian Mathematical Olympiad<br />
139-140<br />
Burgiel, H. 26<br />
butterfly (triangular) 23, 98<br />
Butterfly Map 97-98<br />
Caesariano, Caesare 18-19<br />
Cahill, B. J. S. 97-98<br />
California Institute of Technology 178,<br />
180<br />
Calypso 83<br />
Cambridge University 168, 176, 180<br />
camera obscura 155<br />
Canterbury Puzzles 181<br />
Cantrell, David 45<br />
Cardano, Girolamo 154<br />
cardiac vector 81-82<br />
Carpenter’s Plane 112<br />
cartography vi, 97-98<br />
Case Institute of Technology 180<br />
Castelli, Benedetto 157
Index 223<br />
cat iii, 189<br />
Catholic Church 20<br />
Cauchy, Augustin-Louis 50<br />
Cauchy-Schwarz inequality 169<br />
Cavalieri, Bonaventura 157-158<br />
Cavendish Laboratory 168<br />
CBS Evening News 180<br />
cellular automata 179<br />
Center for New Music and Audio<br />
Technologies (CNMAT) 94<br />
centroid 30, 46<br />
Ceres 164<br />
cevian triangle 38<br />
chainmail (triangular) 7-8<br />
chair (triangular) 195<br />
Chaos Game 59-60<br />
Chartres Cathedral 22<br />
Chebyshev, Pafnuty 165<br />
Chekov, Ensign 27<br />
Chennai Temple 5<br />
Cheops (Khufu) 2<br />
Chess Strategy 181<br />
Chinese Checkers 16<br />
Chinese Remainder Theorem 153<br />
Chinese window lattices 6<br />
Chongsheng Monastery 6<br />
chord progressions 25<br />
Chou Dynasty 6<br />
Christian art 18<br />
Christianity v, 22<br />
Christian mysticism 22<br />
Christ of Saint John of the Cross<br />
17-18<br />
chromaticity 88<br />
Chung, Fan 179<br />
church (triangular) 197<br />
church window (triangular) 196<br />
Chu shih-chieh 59<br />
Ciampoli, Giovanni 157<br />
Cipra, B. 43<br />
circle-of-fifths 25<br />
circumcenter 30, 46<br />
circumcevian triangle 39<br />
circumcircle 30, 52, 69, 76<br />
circumscribing rectangle 47<br />
City College of New York 174<br />
Clark, Andrew 148<br />
classification of cubic curves 160<br />
Cloak of Invisibility 28<br />
closed light paths 54-55<br />
Collège de France 166, 174<br />
College of Invisibles 21<br />
College of San Francesco Saverio 162<br />
Collegio della Sapienza 157<br />
Collegium Carolinum (TU Braunschweig)<br />
163-164<br />
Coltrane, John 25<br />
Combinatory Analysis 171<br />
Companion (Gothic) 20<br />
Computational Confectionery 100<br />
Computer-Generated Encyclopedia of<br />
Euclidean Geometry 38<br />
Condorcet voting 91<br />
Conics 151<br />
constructions v, 30<br />
convex tilings 71<br />
Conway, John Horton 43, 123, 179,<br />
187, 180<br />
Conway’s Soldiers 179<br />
Copernicus 155<br />
Corea, Chick 24<br />
corner triangles 38-39<br />
covering properties 45<br />
coverings v, 45<br />
Coxeter groups 176<br />
Coxeter, H. S. M. 25, 69, 176, 187<br />
Coxeter polytopes 176<br />
CO2 emissions 100<br />
Cratylus 149<br />
Crelle, August 165<br />
Crelle’s Journal 165<br />
crochet model (hyperbolic plane) 75<br />
Cronkite, Walter 180<br />
cross formée 20
224 Index<br />
Cruise, Tom 24<br />
Crusades 20<br />
crystals (triangular) 191<br />
cube 11, 15, 117<br />
cuboctahedron 13<br />
cucumber (triangular) 23<br />
Curiosa Mathematica 147, 169<br />
curves of constant breadth (width)<br />
56<br />
Cyclopedia of Puzzles 181<br />
Dali, Salvador 17, 18<br />
danger sign (triangular) 195<br />
Davenport, Harold 179<br />
Davis, Miles 25<br />
Davis, Philip J. v<br />
De Analysi 160<br />
De Architectura 18<br />
Deathly Hallows 28<br />
Decad 10<br />
deceptahedron 14, 15<br />
Dedekind, Richard 164<br />
De Divina Proportione 11, 153<br />
De Finetti diagram 34, 90<br />
Dekov, D. 38<br />
Delahaye, J.-P. 76<br />
Delahaye product 76<br />
Delenn, Ambassador 27-28<br />
deltahedron 14, 15<br />
Demaine, Erik D. 100<br />
De Moivre, Abraham 125<br />
densest packing of circles 61<br />
Desargues, Girard 159<br />
Descartes, René 11, 21, 148, 151-<br />
152, 156, 161, 184<br />
Descartes’ Rule of Signs 156<br />
diameter of a plane set 73<br />
Dianetics 24<br />
diatonic scale 25<br />
die (triangular) 192<br />
difference triangle 117-118<br />
dihedral group 67<br />
diminished triad 25<br />
Dione 83<br />
Diophantus 153, 157<br />
diploid 90<br />
Dirichlet boundary condition 102<br />
Dirichlet, P. G. Lejeune 167<br />
Dirichlet tessellation 62<br />
Disquisitiones Arithmeticae 164<br />
Dissected Triangle Puzzle 104<br />
dissection into isosceles triangles 107<br />
dissection into similar pieces 108<br />
dissection puzzles vi, 104-108<br />
distances from vertices 40<br />
dodecadeltahedron 15<br />
dodecahedron 15, 117<br />
Dodgson, Charles Lutwidge (Lewis<br />
Carroll) 127, 169, 185<br />
Doolan, E. P. 69<br />
Doomsday algorithm 169<br />
dragons 6<br />
Dreyfus affair 16<br />
ducks (triangular) 189<br />
Dudeney, Henry Ernest 104-105, 181,<br />
187<br />
Duhamel, J. M. C. 166<br />
Duke of Braunschweig 163-164<br />
Dyad 9<br />
Dymaxion Car 176<br />
Dymaxion House 176<br />
Dymaxion Skyocean Projection World<br />
Map 97, 176<br />
Earth 80, 84<br />
ecliptic 84<br />
École des Mines 170<br />
École Normale Supérieure 174<br />
École Polytechnique 166, 170<br />
Edo period 6<br />
effective amount (light) 88<br />
Eggleston, H. G. 45<br />
eheleo funnel 8-9<br />
eigenvalues / eigenfunctions 101
Index 225<br />
Einstein, A. 84, 167, 172<br />
Einthoven’s Triangle (electrocardiography)<br />
81-82<br />
Eisenstein, Gotthold 167<br />
Elder Wand 28<br />
electrical axis (heart) 81-82<br />
electrocardiography vi<br />
electrohydrodynamic (EHD) device<br />
84<br />
electromagnetics 102<br />
electron microscopy 99<br />
Elements iv, 1, 150, 161, 173<br />
ellipsoidal shape 34-35<br />
Emperor Justinian 150<br />
Emperor Rudolph II 155<br />
Eötvös Lorand University (Hungary)<br />
177<br />
equation (equilateral triangle) 31<br />
equiangularity 69<br />
equilateral<br />
shadows 51<br />
spirals 69<br />
triangle i-vii, 1-198<br />
triangles and triangles 43<br />
Equilateral Triangle Method (surveying)<br />
78<br />
Equilateral Triangle Rule (speaker<br />
placement) 92-93<br />
equilateral triangular<br />
anemometer 96<br />
billiards 118-120, 180<br />
fractals 58<br />
lattice 60-61, 120<br />
membrane 100-101<br />
microphone placement 92-93<br />
mosaic 61<br />
prism 100-101<br />
rotor 57<br />
equilic quadrilateral 48<br />
equitriangular unit of area (etu) 33<br />
Erdély, Dániel 128<br />
Erdös-Mordell inequality 53<br />
Erdös-Moser configuration 55<br />
Erdös number 177-178<br />
Erdös, Paul 177-178, 187<br />
Erdös Sphere Coloring Problem 110-<br />
111<br />
error-correcting code 91-92<br />
ESA 84<br />
Escher, Maurits 176, 194<br />
Essay on Conic Sections 159<br />
Essay on the Foundations of Geometry<br />
173<br />
Eternity Puzzle 124-125<br />
ETH-Zurich 169, 172<br />
Euclidean geometry 126-127<br />
Euclid of Alexandria iv, 1, 30, 150-<br />
152, 154, 158, 183<br />
Eudoxus 149-150<br />
Euler, Leonhard 11, 125, 161, 184<br />
Euler line 46, 161<br />
Euler’s inequality 52, 161<br />
Eves, Howard 66<br />
excenter 46<br />
excentral triangle 46<br />
excentral triangle iteration 46<br />
Eye of God 18<br />
Fagnano, Giovanni 54<br />
Fagnano, Giulio 54<br />
Fagnano orbit 118-120<br />
Fagnano Problem 54, 119, 170<br />
fallout shelter (triangular) 196<br />
Father 18<br />
Fejér, L. 177<br />
Fermat, Pierre de 35, 50, 153, 156,<br />
159, 184<br />
Fermat point 35-36, 39, 158<br />
Fermat prime 164<br />
Fermat’s Factorization Method 157<br />
Fermat’s Last Theorem 157, 161<br />
Fermat’s Little Theorem 157<br />
Fermat’s Method of Descent 157<br />
Fermat’s Polygonal Number Conjec-
226 Index<br />
ture 157<br />
Fermat’s Principle of Least Time 157<br />
Fermat-Torricelli Problem 35, 158-<br />
159<br />
Feynman, Richard P. 109<br />
Fibonacci numbers 67, 69, 153<br />
Fibonacci sequence 67, 70<br />
Fibonacci Triangle 67-68<br />
Fields Medal 151<br />
fighters (triangular) 189<br />
figurate numbers 32<br />
Finite Semifields and Projective Planes<br />
180<br />
Fisher, Irving 97-99<br />
Flag of Israel 16<br />
flag (triangular) 193<br />
flammability diagram 86-87<br />
flammability region 86<br />
flexagon 108-109<br />
Flexagon Committee 109<br />
Flint, J. A. 81<br />
Flint, Michigan 151<br />
Florentine enigma 159<br />
flower (triangular) 23<br />
Fludd, Robert 9<br />
fly (triangular) 23<br />
Fontana, Niccolò (Tartaglia) 154, 120,<br />
183<br />
Foucault, Léon 158<br />
four elements 9-11, 24<br />
Four Squares Theorem 163<br />
Fourth Dynasty 2<br />
Four Triangle Sculpture 25<br />
fractal dimension 58<br />
Fraenkel, A. S. 123<br />
Freemasonry 20-21<br />
Free Will Theorem 169<br />
French Mathematical Olympiad 140<br />
Freudenthal, H. 15<br />
Friedman, Erich 44-45<br />
Frost, E. 15<br />
fullerenes 176<br />
Fuller, R. Buckminster 97-99, 176,<br />
186<br />
fundamental domain 67<br />
fundamental mode 101<br />
Fundamental Theorem of Algebra 164<br />
fundamental triangle 90<br />
Fundamenta Mathematica 175<br />
Fürst, Paul 100<br />
Galileo 157-158<br />
Gallery of Equilateral Triangles vi,<br />
26, 188-198<br />
Game of Life 169<br />
Gamesters of Triskelion 27<br />
game theory vi, 90<br />
game (triangular) 192<br />
Gardner, Martin vi, 43, 119, 169,<br />
179, 181-182, 187<br />
Gardner’s Gaffe 119-120<br />
Garfunkel-Bankoff inequality 53<br />
Gateway Arch 26-27<br />
Gaussian curvature 164<br />
Gauss, Karl Friedrich 50, 163, 167,<br />
185<br />
Gauss Plane 50, 164<br />
Gauss’ Theorem on Triangular Numbers<br />
50, 164<br />
Gazalé, M. J. 69<br />
geese (triangular) 189<br />
gem (triangular) 191<br />
General Theory of Relativity 84, 167,<br />
172<br />
generating parallelogram 61<br />
genetics vi, 90<br />
genotype 90<br />
geodesic dome 99, 176, 195<br />
geometric spiral 69<br />
Géométrie 156<br />
Geometrie der Zahlen 172<br />
Geometry v<br />
Gerbert of Aurillac v<br />
Gergonne point 39
Index 227<br />
Global Positioning System 80<br />
global warming 100<br />
gnana 5<br />
gnomon 70<br />
Goethe color triangle 86-87<br />
Goldberg, Michael 66, 107<br />
golden<br />
mean 2<br />
ratio 68-69<br />
section 13, 58, 153<br />
Golden Triangle Fractal 58<br />
Golomb, Solomon Wolf 112, 178, 187<br />
Gonville and Caius College, Cambridge<br />
179<br />
Gordon, J. M. 94<br />
Gothic cathedral 18-19, 22<br />
Gothic masons 20<br />
GPS antenna 79-80<br />
Graham, Ronald Lewis 73, 177-178,<br />
187<br />
Grand Architect of the Universe 21<br />
Grand Duke Ferdinando II de’ Medici<br />
157-158<br />
Grandi, Luigi Guido 159<br />
gravitational waves vi, 84<br />
Great Pyramid (Giza) 2, 3<br />
Greek Symbol Puzzle 103, 181<br />
Greenhill, Alfred George 171<br />
green pepper (triangular) 23<br />
Green, Trevor 45<br />
group of isometries 67<br />
Grunsky, H. 77<br />
Guinness Book of Records 178<br />
Haberdasher’s Puzzle 104<br />
Hales, Thomas 155<br />
Halley, Edmond 162<br />
Hall, Jr., Marshall 180<br />
Halma 16<br />
Hamada, N. 92<br />
Hamlyn, P. 72<br />
Hamming<br />
code 92<br />
distance 92<br />
hana gusari 7-9<br />
handshake problem 32<br />
Harboth, Heiko 125<br />
Hardy-Weinberg frequency 90<br />
Harmonice Mundi 15, 155<br />
Harriot, Thomas 148<br />
Harvard University 176, 178<br />
Hasse, Helmut 175<br />
hat (triangular) 193<br />
Haverford College 171<br />
Haydn, Franz Joseph 25<br />
heat transfer 102<br />
heccaidecadeltahedron 15<br />
Hecke, Erich 175<br />
Heiberg, J. L. 151<br />
Helene 83<br />
heliotrope 164<br />
Hellenic Mathematics 150<br />
Heraclitus 149<br />
Heraldic Cross 20<br />
Hermite, Charles 166, 170<br />
Hertz, Heinrich 169<br />
hexagon 10, 16, 107<br />
hexagram 10, 16<br />
hexiamonds 112, 112-114, 178<br />
Hilbert, David 172<br />
Hilbert’s Eighth Problem 167<br />
Hilbert’s Problems 172<br />
Hinduism v, 4, 5, 16<br />
hinged dissection 104-105<br />
Hioka, Y. 92<br />
history (equilateral triangle) v, 1-28<br />
hollow triangle 26<br />
Holy Lands 20<br />
Holy Spirit 18<br />
homogeneous coordinates 33-34<br />
honeydew melon (triangular) 23<br />
Hooke, Robert 160<br />
Hop Ching Checkers 16<br />
Hopkins, A. A. 30
228 Index<br />
house (triangular) 197<br />
Hubbard, L. Ron 24<br />
Hudson River Psychiatric Center 69<br />
hue 88<br />
human elbow 82<br />
humor (triangular) 188<br />
Humpty Dumpty Magazine 182<br />
Hungarian National Olympiad 141<br />
Hurwitz, Adolf 172<br />
Huyghens, Christiaan 160<br />
hyperbolic equilateral triangles 74<br />
hyperbolic plane<br />
(models) 75<br />
(standard) 74<br />
Iberoamerican Mathematical Olympiad<br />
(Mexico) 142<br />
iccha 5<br />
Ice Age 2<br />
icosahedral speaker 94<br />
icosahedral symmetry 99<br />
icosahedron 11, 15, 97-99, 117<br />
icosidodecahedron 13<br />
ida 5<br />
Ignis 9<br />
impossible triangle 196<br />
imputation vector 90<br />
incenter 30, 39<br />
incircle 30, 46, 56<br />
incircle-triangle iteration 46<br />
infinitely many linear equations 173<br />
inorganic chemistry 24<br />
Intégrale, longeur, aire 174<br />
International Juggler’s Association<br />
179<br />
International Mathematical Olympiad<br />
129, 137-138, 143<br />
Introduction to Geometry 176<br />
Intuition 25-26<br />
ionic wind 85<br />
ionocraft 84-85<br />
Irish Mathematical Olympiad 139,<br />
144<br />
Iron Cross 20<br />
isodynamic points 38-39<br />
isogonic center 35-36<br />
isometry 75<br />
Isoperimetric Theorem 52<br />
Jacobi, C. G. 50, 165, 167<br />
Jagiellonian University (Kraków) 175<br />
Jansenism 159<br />
Japanese Temple Geometry 126<br />
Jefferson National Expansion Memorial<br />
26<br />
Jerrard, R. P. 43<br />
Jesuit College of La Flèche 156<br />
Jesus Christ 18, 22<br />
Jet Propulsion Laboratory 178<br />
Johns Hopkins University 171, 178<br />
Johnson, R. S. 107<br />
Judaism v, 16<br />
Jung’s Theorem 52<br />
Jupiter 83<br />
Kabbalah 10, 16<br />
Kabun 8<br />
Kakeya Needle Problem 57-58<br />
Kali (Kalika) 5<br />
Kelly, L. M. 75<br />
Kepler Conjecture 155<br />
Keplerian telescope 155<br />
Kepler, Johannes 15, 21, 154, 184<br />
Kepler’s Laws 155<br />
Kepler’s solids 155<br />
Kepler’s supernova 155<br />
Khufu (Cheops) 2, 3<br />
King David 16<br />
King Hieron 151<br />
King Ptolemy 150<br />
King’s College (Cambridge) 171<br />
King’s College (London) 168<br />
Kirk, Captain 27<br />
Kline, Morris 153
Index 229<br />
Knight’s Templar 20<br />
Knight’s Tours 125, 161<br />
Knuth, Donald Ervin 119-120, 180,<br />
187<br />
Kochen, S. B. 169<br />
Koch, Helge von 173, 186<br />
Koch Snowflake / Anti-Snowflake 58,<br />
173<br />
Korean Mathematical Olympiad 139<br />
KRC triangle 24<br />
kriya 5<br />
Kummer, Ernst 169<br />
kusari 8<br />
Lagrange, Joseph-Louis 50, 162, 185<br />
Lagrange multipliers 162<br />
Lagrange’s Equilateral Triangle Solution<br />
83<br />
Lagrangian function 163<br />
Lagrangian (L-, libration) points 83<br />
Lamé, Gabriel 102<br />
Lamp 112<br />
Laplace operator 101<br />
Laplace, Pierre-Simon 161<br />
Laplacian eigenstructure 101<br />
Large Hadron Collider 84<br />
largest<br />
circumscribed equilateral triangle<br />
36<br />
inscribed rectangle 41<br />
inscribed square 41<br />
inscribed triangle 52<br />
Lasker, Emmanuel 172<br />
Last Supper 13, 17, 18<br />
lateral epicondoyle 82<br />
lattice duality 62<br />
Latvian Mathematical Olympiad 144<br />
least-area rotor 56-57<br />
least-diameter decomposition (open<br />
disk) 52<br />
Lebesgue constant 174<br />
Lebesgue covering dimension 174<br />
Lebesgue, Henri 174, 186<br />
Lebesgue integral 174<br />
Lebesgue measure 174<br />
Lebesgue’s Dominated Convergence<br />
Theorem 174<br />
Lebesgue spine 174<br />
Lebesgue-Stieltjes integral 174<br />
Leech lattice 169<br />
Legendre, Adrien-Marie 125-126<br />
Lehmer, D. H. 178<br />
Leibniz, Gottfried 156, 159-160<br />
Leonardo of Pisa (Fibonacci) 152,<br />
183<br />
Leonardo’s Symmetry Theorem 154<br />
Lepenski Vir 1-2<br />
Letters to a Princess of Germany<br />
161<br />
Leunberger inequality (improved) 53<br />
Liber Abaci (Book of Calculation)<br />
153<br />
Library of Alexandria 150<br />
Lifter 84-85<br />
Likeaglobe Map 97-99<br />
limiting oxygen concentration (LOC)<br />
86<br />
Lindgren, Harry 107<br />
LISA 84<br />
Liszt, Franz 25<br />
Lob, H. 66<br />
lodge (triangular) 197<br />
logarithmic spiral 70<br />
London Bridge Station 86<br />
Los, G. A. 66<br />
lower explosive limit (LEL) 86<br />
Loyd, Sam 103, 181, 187<br />
Lucasian Chair 160<br />
lunar crater (triangular) 191<br />
Lux 9<br />
Luzin, Nikolai 175<br />
Machine for Questions and Answers<br />
38
230 Index<br />
MacMahon, Percy Alexander 113,<br />
171, 186<br />
MacMahon’s Color Triangles 113, 115-<br />
116<br />
MacTutor History of Mathematics<br />
vi<br />
magic v, 16<br />
constant 16<br />
hexagram 16<br />
number 16<br />
star 16<br />
major third 25<br />
major triad 25<br />
Makhuwa 8<br />
Malfatti circles 66<br />
Malfatti, Gian Francesco 65, 162,<br />
184<br />
Malfatti resolvent 162<br />
Malfatti’s Problem 65-66, 162, 166<br />
Maltese cross 20<br />
Manchester University 177<br />
mandorla 22<br />
Manfredi, Gabriele 162<br />
mantra 5<br />
mappings preserving equilateral triangles<br />
75<br />
marble problem 65<br />
Marischal College (Aberdeen) 168<br />
Mars 155<br />
Marshall, W. 72<br />
Martin Co., Glenn L. 178<br />
Marundheeswarar Temple 5<br />
Masonic Lodges 21<br />
Masonic Royal Arch Jewel 21<br />
Mästlin, Michael 155<br />
Mathemagician 182<br />
mathematical<br />
biography v<br />
competitions v-vi, 129-147<br />
history v<br />
properties v, 29-77<br />
recreations v, 103-128<br />
Mathematical Association of America<br />
11, 162, 169, 182<br />
Mathematical Games 119, 182<br />
Mathematical Olympiad of<br />
Moldova 145<br />
Republic of China 142<br />
Mathematical Recreations and Essays<br />
176<br />
Mathematical Snapshots 105<br />
Mathematical Time Exposures 105<br />
matrix representation 91-92<br />
maximum area rectangle 41<br />
Maxwell-Boltzmann kinetic theory<br />
of gases 168<br />
Maxwell color triangle 88, 168<br />
Maxwell discs 168<br />
Maxwell distribution 169<br />
Maxwell, James Clerk 167, 185<br />
Maxwell materials 169<br />
Maxwell model (generalized) 169<br />
Maxwell’s demon 169<br />
Maxwell’s equations 168-169<br />
Maxwell’s Theorem 169<br />
McCartin, Brian J. 102<br />
McGovern, W. 15<br />
Mécanique analytique 163<br />
Mechanica 161<br />
medial epicondoyle 82<br />
medial triangle 54<br />
Melissen, J. B. M. 44-45<br />
Mersenne, Marin 156-157, 159<br />
Mesolithic 1-2<br />
METAFONT 180<br />
method of<br />
exhaustion 150<br />
indivisibles 157<br />
Méthodes nouvelles de la mécanique<br />
céleste 170<br />
Method of Mechanical Theorems 151<br />
Meyer Sound 94<br />
Middle Ages 22, 121, 152<br />
midpoint ellipse 51
Index 231<br />
Milan Cathedral 18-19<br />
Millenium Prize Problems 167<br />
minimum order perfect triangulation<br />
72<br />
Minkowski, Hermann 172, 186<br />
Minkowski plane 75, 173<br />
Minkowski spacetime 172<br />
minor triad 25<br />
Miscellaneous Olympiad Problems 146-<br />
147<br />
Mittag-Leffler, Gösta 173<br />
M-matrices 173<br />
Möbius band 194<br />
Modern Puzzles 181<br />
Monad 9<br />
Monckton, Christopher 125<br />
Morandi, Maurizio 44<br />
Morgenstern, Oskar 90<br />
Morley, Frank 171, 186<br />
Morley’s Theorem v, 35, 171-172<br />
Morley triangle 35<br />
Moslems 20<br />
Most Beautiful Theorems in Mathematics<br />
61<br />
MOTET v-vi<br />
moth (triangular) 190<br />
Motzkin, T. 77<br />
Mozartkugel 100<br />
Mozart, Wolfgang Amadeus 21, 100<br />
music v, 25<br />
musical<br />
chord 25<br />
harmony 10<br />
triangle 192<br />
Musselman, J. R. 39<br />
Musselman’s Theorem 39<br />
nadis 5<br />
Nambokucho period 8<br />
Napoleon’s Theorem v, 36-37<br />
Napoleon triangle 36, 39<br />
NASA 84<br />
NASA Echo I 80<br />
natural equilateral triangles 97-98<br />
Nazi Party 175<br />
nearly-equilateral dissections 73<br />
Nelson, H. L. 107<br />
neopythagoreans 10<br />
Neumann boundary condition 102<br />
neutron star 84<br />
Neville truss 86<br />
New Mathematical Pastimes 171<br />
Newton form 160<br />
Newton’s identities 160<br />
Newton, Sir Isaac 1, 24, 152, 156,<br />
159-161, 184<br />
Newton’s law of cooling 160<br />
Newton’s Laws 160<br />
Newton’s method 160<br />
nine-point center 19, 46<br />
no equilateral triangles on a chess<br />
board 61<br />
nonagon (enneagon) 107<br />
nonattacking rooks 126<br />
non-Euclidean equilateral triangles<br />
74<br />
Nordic Mathematics Competition 144<br />
North American Network 79-80<br />
O’Beirne, T. H. 112<br />
octahedral group 26<br />
octahedron 11, 15, 97-98, 117<br />
Odom, George 25-26, 69, 177<br />
olecranon 82<br />
olympiad problems vi<br />
On Finite Sums of Rational Numbers<br />
178<br />
On the hypotheses that lie at the foundations<br />
of geometry 167<br />
On the number of primes less than<br />
a given magnitude 167<br />
optimal spacing (lunar bases) 111<br />
optimal wrapping 100<br />
Order of the Temple 20
232 Index<br />
oriented triangles 70<br />
origami 127<br />
orthic triangle 54<br />
orthocenter 39, 46<br />
Ostrowski, Alexander 173<br />
Oughtred, William 148<br />
Our Lord 18<br />
Oxford University 150<br />
Pach, J. 55<br />
Pacioli, Luca 11, 153<br />
packings v, 44<br />
Padovan sequence 70<br />
Padovan spirals 70<br />
Padovan whorl 70<br />
pagoda 6-7<br />
paper folding 127<br />
palimpsest 151<br />
Pappus’ Centroid Theorem 152<br />
Pappus’ Hexagon Theorem 152<br />
Pappus of Alexandria 13, 151-152,<br />
183<br />
Pappus’ Problem 152<br />
parallelogram properties 37<br />
Parthenon 9<br />
partition 72-73<br />
partridge<br />
number 72<br />
tiling v, 72<br />
Pascal, Blaise 58, 157, 159, 184<br />
Pascal’s Law of Pressure 159<br />
Pascal’s Mystic Hexagram Theorem<br />
159<br />
Pascal’s Triangle 58-59, 74, 159<br />
Pascal’s Wager 159<br />
Pauca sed matura 164<br />
pedal mapping 54<br />
pedal triangle 38-39, 54, 118, 127<br />
Pedersen, Jean J. 117<br />
Pedoe, Dan 31<br />
peg solitaire 169<br />
Pell, John 148<br />
Pell’s equation 157<br />
Peloponnesian War 149<br />
pengala 5<br />
pentagon 107<br />
pentagonal dipyramid 15<br />
Perelman, Grigory 170<br />
perfect<br />
coloring 67<br />
tiling 70<br />
triangulation 70<br />
periodic orbit 118-120<br />
Perplexities 181<br />
Pestalozzi, J. H. 165<br />
Peterhouse, Cambridge 168<br />
Pfaff, J. F. 164<br />
Philosophia Sacra 9<br />
Picard, Émile 166<br />
Pinchasi, R. 55<br />
ping-pong 178<br />
plaited polyhedra 117<br />
planar soap bubble clusters 52<br />
plastic<br />
number 70<br />
pentagon 70<br />
Platonic solid 11-13, 117, 150<br />
Platonic triangles 11<br />
Platonism 150<br />
Plato of Athens 11, 149, 183<br />
Plato’s Academy 149-150<br />
playscape (triangular) 195<br />
Poincaré-Bendixson Theorem 170<br />
Poincaré Conjecture 170<br />
Poincaré disk 75<br />
Poincaré group 170<br />
Poincaré, Henri 166, 170, 185<br />
Poincaré inequality 170<br />
Poincaré-Linstedt method 170<br />
Poincaré map 170<br />
Poincaré metric 170<br />
Poincaré, Raymond 170<br />
Polydeuces 83<br />
polygonal dissections 106-107
Index 233<br />
Polygonal Number Theorem 50, 163<br />
polyhedral formula 11, 156, 161<br />
polyhedron 11, 13, 15<br />
Polyominoes 178<br />
Pompeiu’s Theorem 49-50<br />
Poncelet-Steiner Theorem 165<br />
Pontormo, Jacopo 17-18<br />
pool balls (triangular) 192<br />
pool-ball triangle 117-118<br />
positively / negatively equilateral triangle<br />
39-40<br />
Post, K. 43<br />
Post’s Theorem 43<br />
Potter, Harry 28<br />
Precious Mirror of the Four Elements<br />
59<br />
Pressman & Co., J. 16<br />
primary color 86<br />
Prime Number Theorem 173, 177<br />
Princeps mathematicorum 163<br />
Princeton University 109, 176-177,<br />
180<br />
principal frequency 100<br />
Principia Mathematica (Newton) 1,<br />
160-161<br />
Principia Mathematica (Whitehead<br />
& Russell) 173<br />
Principle of Least Action 162<br />
Principle of the Equilateral Triangle<br />
(electrocardiography) 81<br />
Principle of Virtual Work 163<br />
Problem of Apollonius 151-152<br />
Problems in the Distribution of Prime<br />
Numbers 178<br />
projective plane 91-92<br />
Propeller Theorem 64<br />
Ptolemy 152<br />
pumpkin (triangular) 193<br />
Puzzles and Curious Problems 181<br />
pyramid 2-3<br />
Pythagoras of Samos 149, 183<br />
Pythagoreans 9, 22, 149<br />
Pythagorean Theorem 149, 153<br />
QFL diagram 88-89<br />
Qianxun Pagoda 6<br />
Qing Dynasty 6<br />
quadratix of Hippias 152<br />
Quan Fengyou, King 6<br />
quantum mechanics 102<br />
Queen Anne 160<br />
Queen Christina 156<br />
Queen of the Sciences 164<br />
quilt (triangular) 194<br />
Rabbinic Judaism 16<br />
random point 50<br />
ranking region 91<br />
rational triangle 48<br />
Ravensburger Spieleverlag GmbH 16<br />
Recreational Mathematics vi, 103,<br />
178, 182<br />
regular<br />
heptadecagon 164<br />
polytope 73-74<br />
rhombus 61<br />
simplex 73<br />
tessellations (planar) 61<br />
tilings 15, 156<br />
Renaissance 18, 22<br />
representation triangle 91<br />
rep-tiles vi, 112-113, 178<br />
restricted three-body problem 83<br />
Resurrection 18<br />
Resurrection Stone 28<br />
Reuleaux Triangle 55-56, 96<br />
Ricatti, Vincenzo 162<br />
Richmond, H. W. 66<br />
Riemann, Bernhard 164, 166, 185<br />
Riemann Hypothesis 167, 173<br />
Riemannian metric 167<br />
Riemannian space 167<br />
Riemann integrability 167<br />
Riemann-Lebesgue Lemma 174
234 Index<br />
Riemann Surfaces 51, 167<br />
Riemann zeta function 167<br />
Riordan, Oliver 124-125<br />
Ripley’s Believe It or Not 179<br />
Roberval, G. P. de 159<br />
Robin boundary condition 102<br />
Robinson, John 25-26<br />
Robinson, J. T. R. 96<br />
rock formation (triangular) 191<br />
Romanian IMO Selection Test 140<br />
Romanian Mathematical Olympiad<br />
140<br />
root movement 25<br />
Rosicrucian cross 21<br />
Rosicrucians 21, 156<br />
Royal Artillery School (Turin) 162<br />
Royal Institute of Technology (Stockholm)<br />
173<br />
Royal Military Academy (Woolwich)<br />
171<br />
Royal Society 105<br />
Rubik, Ernö 128<br />
Rubik’s Cube 128<br />
Rudolphine Tables 155<br />
Russell, Bertrand 1, 169, 173, 186<br />
Russell’s Paradox 173<br />
Russian Mathematical Olympiad 140<br />
rusty compass construction 30-31,<br />
153<br />
Saari, D. G. 91<br />
Sacred Geometry 22, 69<br />
sacred paintings 18<br />
Saint John of the Cross 18<br />
sakthis 5<br />
Sakuma, Tumugu 8<br />
salt 9<br />
Sangaku Geometry 8, 126<br />
Santa Clara University 117<br />
Santa Maria delle Grazie 18<br />
satellite geodesy 79-80<br />
saturation 88<br />
Saturn 83<br />
Saturn’s rings 168<br />
Saviour 18<br />
Schellbach, K. H. 66<br />
Schoenberg, I. J. 77, 105<br />
Schubert, Franz 25<br />
Schwarz alternating method 169<br />
Schwarz-Christoffel formula 169<br />
Schwarz, Hermann Amandus 54, 169,<br />
185<br />
Schwarzian derivative 169<br />
Schwarz minimal surface 169<br />
Schwarz reflection principle / procedure<br />
54, 119, 169<br />
Schwarz’s Lemma 169<br />
Scientific American 119, 178, 181-<br />
182<br />
Scientology v, 24<br />
Scott, C. 80<br />
secondary color 86<br />
Second Punic War 151<br />
Selberg, Atle 177<br />
Selby, Alex 124-125<br />
semiregular tilings 15, 156<br />
Seversky, A. P. de 84<br />
Sforza, Ludovico (Duke of Milan) 11<br />
shawl (triangular) 194<br />
Sheridan, Captain 27<br />
Sherlock Holmes 181<br />
Shield of David 16<br />
shift register sequences 178<br />
Shinto 126<br />
Shiva 5<br />
shogun 126<br />
shortest bisecting path 53<br />
Sicherman, George 117<br />
siege of Syracuse 151<br />
Sierpinski carpet 175<br />
Sierpinski curve 175<br />
Sierpinski Gasket 58-59, 175<br />
Sierpinski number 175<br />
Sierpinski Problem 175
Index 235<br />
Sierpinski, Waclaw 174, 186<br />
Sikorska, J. 75<br />
Silver Age of Greek Mathematics 152<br />
similarity transformation 75<br />
Simmons, G. J. 111<br />
simplex plot 34<br />
Sinclair, Commander 27-28<br />
Singmaster, David 122-123<br />
six triangles 48-49<br />
smallest<br />
enclosing circle 52<br />
inscribed triangle 54<br />
small rhombicosidodecahedron 13<br />
small rhombicuboctahedron 13<br />
Smith’s Prize 168<br />
Snail 112<br />
Snefru 2<br />
Snell’s Law 156-157<br />
snow crystal (triangular) 98<br />
snub cube 13<br />
snub dodecahedron 13<br />
Society of Apostles 168, 173<br />
Socrates 149<br />
Socratic dialogues 149<br />
so gusari 8<br />
Soma cube 179<br />
Son 18<br />
Sorbonne 174<br />
Southern Illinois University 176<br />
Spaceship Earth 176<br />
space-time continuum 172<br />
Special Theory of Relativity 172<br />
spherical equilateral triangles 74<br />
Sphinx 112<br />
spidrons vi, 127-128<br />
Square and Triangle Puzzle 105-106<br />
square hole drill vi, 95-96<br />
stamp (triangular) 192<br />
Stanford University 180<br />
Star of David 16<br />
Star Trek 27<br />
Steiner ellipses 166<br />
Steiner, Jakob 66, 165, 167, 175, 185<br />
Steiner surface 165<br />
Steiner Theorem 165<br />
Steiner triple-system 91, 166<br />
Steinhaus, Hugo 43, 105<br />
Stephen the Clerk 150<br />
Stern-Halma 16<br />
Stertenbrink, Günter 124-125<br />
St. John’s College, Cambridge 168<br />
Stockholm University 173<br />
Stone Age 2<br />
Stone, Arthur H. 109<br />
stone (triangular) 191<br />
St. Petersburg Academy of Sciences<br />
161<br />
Strand Magazine 181<br />
Straus, E. G. 111<br />
street signs (triangular) 196<br />
strictly convex position 55<br />
Sun 83-84<br />
superconducting (Sierpinski) gasket<br />
vi, 94-95<br />
Supper at Emmaus 17, 18<br />
Supreme Fascist 177<br />
surreal numbers 179-180<br />
sushumna 5<br />
Suzuki, Fukuzo 8<br />
sweets (triangular) 193<br />
Symbolic Logic 169<br />
symmetry<br />
group 25, 66<br />
tiling 67<br />
Synagoge (Collection) 152<br />
Synergetics 176<br />
syzygies 42-43<br />
Szostok, T. 75<br />
Tahiti Triangle 190<br />
Tang Dynasty 6<br />
Tangencies 151<br />
tantra 5<br />
Tarot 10
236 Index<br />
Tartaglian Measuring Puzzles 120-<br />
121<br />
tattvas 5<br />
Tatzenkreuz 20<br />
Technical University of Berlin 169<br />
Technical University of Braunschweig<br />
125<br />
Technische Hochschule (Graz) 175<br />
telegraph 164<br />
Telesto 83<br />
Tenebrae 9<br />
Tercentenary Euler Celebration 162<br />
ternary diagram 33-34, 88, 90, 121<br />
Terra 9<br />
tertiary color 86<br />
tessellations to fractals 62-63<br />
Tethys 83<br />
tetracaidecadeltahedron 15<br />
Tetragrammaton 10<br />
tetrahedral geodesics 64-65<br />
tetrahedron 9, 11, 15, 117<br />
tetraktys 9-10<br />
Tetris 178<br />
TeX 180<br />
Thales 149<br />
Theaetetus 149-150<br />
Theobald, Gavin 107<br />
Theodorus 149<br />
Theorema Egregium 164<br />
Theoria motus corporum coelestium<br />
164<br />
Theory of Forms 149-150<br />
Thirty Years’ War 156<br />
three-body problem 83, 163<br />
three jug problem 120-121, 154<br />
Three Pagodas 6-7<br />
Thurston model 75<br />
Timaeus 11, 150<br />
Torricelli, Evangelista 35, 157-158,<br />
184<br />
Torricelli’s Equation 158<br />
Torricelli’s Law / Theorem 158<br />
Torricelli’s Trumpet (Gabriel’s Horn)<br />
158<br />
torsional rigidity 100<br />
Traffic Jam Game 123<br />
Traité du triangle arithmétique 58,<br />
159<br />
Trattato della Pittura 153<br />
Trattato generale di numeri e misure<br />
154<br />
Travolta, John 24<br />
Treatise on Electricity and Magnetism<br />
168-169<br />
triangle 9<br />
Triangle and Square Puzzle 105<br />
Triangle Book 180<br />
triangle geometry v<br />
Triangle Inequality 52<br />
triangle in rectangle 41<br />
triangle in square 41-42<br />
triangles with integer sides 122-123<br />
Triangle-to-Triangle Dissection 106<br />
Triangle Waist Co. 198<br />
triangular<br />
bounds 100-101<br />
dipyramid 15<br />
honeycomb 125-126<br />
numbers 10, 32-33, 59<br />
triangulation triangles 38-39<br />
trihexaflexagon 108-109<br />
trilinear coordinates 33<br />
Triluminary 27-28<br />
Trinity 4, 18<br />
Trinity College, Cambridge 160, 173<br />
Tripurasundari 5<br />
trisection thru bisection 32<br />
Trojan asteroid 83<br />
truncated<br />
cube 13<br />
dodecahedron 13<br />
octahedron 13<br />
tetrahedron 13<br />
Tschirnhaus, E. W. von 159
Index 237<br />
Tuckerman, Bryant 109<br />
Tukey, John W. 109<br />
tulip (triangular) 189<br />
Turkish Mathematical Olympiad 141<br />
Tutte, W. T. 70<br />
Tweedie, M. C. K. 120<br />
two-color map 110<br />
UFO (triangular) 190<br />
Uhura, Lieutenant 27<br />
Ulam, Stanislaw 177<br />
Umble, R. 120<br />
University of Alaska 178<br />
University of Basel 161<br />
University of Berlin 165-166, 169<br />
University of Bonn 172, 175<br />
University of Breifswald 175<br />
University of Caen 170<br />
University of California-Berkeley 94,<br />
178<br />
University of California-San Diego<br />
179<br />
University of Chicago 178, 182<br />
University of Ferrara 162<br />
University of Göttingen 164, 166,<br />
169, 172, 175<br />
University of Halle 169<br />
University of Hamburg 175<br />
University of Helmstedt 164<br />
University of Königsberg 172, 175<br />
University of Lvov 175<br />
University of Orleans 157<br />
University of Paris 170<br />
University of Pisa 175<br />
University of Poitiers 156<br />
University of Prague 175<br />
University of Southern California 178<br />
University of Toronto 176<br />
University of Tübingen 155, 175<br />
University of Vienna 175<br />
University of Warsaw 175<br />
upper explosive limit (UEL) 86<br />
USA Mathematical Olympiad 129,<br />
136-137<br />
U. S. Chess Hall of Fame 181<br />
U. S. Coast and Geodetic Survey 80<br />
USDA Soil Texture Triangle 88-89<br />
Valen 27-28<br />
Van der Waerden, B. L. 15<br />
ventricular hypertrophy 81<br />
vesica piscis 22<br />
vibration theory 102<br />
Victoria cross 20<br />
da Vinci, Leonardo 11, 13, 17-18,<br />
153, 183<br />
Virgin Mary 22<br />
viruses 99<br />
Vishnavite 16<br />
Vishnu 5, 16<br />
Vitruvius 18<br />
Viviani’s Theorem 33-34<br />
Viviani, Vincenzo 157-158, 184<br />
Viviani Window 159<br />
Von Neumann, John 90<br />
Voronoi diagram 62<br />
Voronoi, G. F. 175<br />
voting<br />
paradox 91<br />
theory vi, 91<br />
Voyager spacecrafts 168<br />
Wackhenfels, Baron von 155<br />
Wagner, Richard 25<br />
Wagner, Rudolf 165<br />
Wainwright, R. T. 71<br />
Wallenstein, General 155<br />
Wallis, John 76, 148<br />
Wanderer Fantasy 25<br />
Wankel engine 96<br />
Warren truss 85-86<br />
Wasan 7<br />
Washington, George 21<br />
Watts drill 96
238 Index<br />
Weber, Wilhelm 164<br />
Weierstrass, Karl 169<br />
Weisstein, E. W. 30<br />
Weizmann Institute of Science 123<br />
Wetzel, J. E. 41, 43<br />
whorled equilateral triangle 69<br />
Widder, D. V. 178<br />
Wikipedia, The Free Encyclopedia vi<br />
Wiles, Andrew 157<br />
Wilson’s Theorem 163<br />
wing (triangular) 190<br />
Winter, Edgar 24<br />
Winter Triangle 190<br />
Wirtinger, Wilhelm 175<br />
World Chess Champion 172<br />
World War I 175<br />
World War II 175, 182<br />
Wrangler (Tripos) 168<br />
wrapping chocolates vi<br />
yantra 5, 16<br />
Zalgaller, V. A. 66<br />
Zanotti, Francesco Maria 162<br />
Zaremba, S. 175<br />
Zathras 27<br />
zeta function 161<br />
Ziegler’s Giant Bar 180<br />
Zionism 16